Additive averages of multiplicative correlation sequences and applications
AADDITIVE AVERAGES OF MULTIPLICATIVE CORRELATIONSEQUENCES AND APPLICATIONS
SEBASTIÁN DONOSO, ANH N. LE, JOEL MOREIRA, AND WENBO SUN
Abstract.
We study sets of recurrence, in both measurable and topological settings, foractions of ( N , × ) and ( Q > , × ) . In particular, we show that autocorrelation sequences ofpositive functions arising from multiplicative systems have positive additive averages. Wealso give criteria for when sets of the form { ( an + b ) (cid:96) / ( cn + d ) (cid:96) : n ∈ N } are sets ofmultiplicative recurrence, and consequently we recover two recent results in number theoryregarding completely multiplicative functions and the Omega function. Introduction and results
An equation of several variables is partition regular if for any finite coloring of N , thereexists a solution to the equation with all variables having the same color. An old question ofErdős and Graham [10] asks whether the equation x + y = z is partition regular. A partialanswer was obtained in 2016, when the conjecture was confirmed in the case of two colors[15]. In general, the answer to Erdős and Graham’s question is still unknown even if we justrequire two of the variables to have the same color. Conjecture 1.1.
For any finite coloring of N , there exist x, y of the same color such that x + y (or x − y ) is a perfect square. Frantzikinakis and Host proved in [11] that for any finite coloring of N , there are x, y ofthe same color such that x + 9 y is a perfect square. Expanding on these ideas, the fourthauthor established an analogue of Conjecture 1.1 where N is replaced with the ring of integersof a larger number field (see [22, 23]). However the methods used there do not seem to applyto N .The above results were proved by first recasting the combinatorial problems into questionsin ergodic theory, in particular about sets of return times in multiplicative measure preservingsystems. For a semigroup G , a topological G -system is a pair ( X, T ) where X is a compactmetric space and T is a G -action on X , i.e. for g ∈ G , T g : X → X is a homeomorphism andfor g, h ∈ G , T g ◦ T h = T gh . Let B be the Borel σ -algebra and µ be a probability measure on X such that µ ( T − g A ) = µ ( A ) for all A ∈ B . Then the tuple ( X, B , µ, T ) is called a measure Mathematics Subject Classification.
Primary: 37B20, 37A45 ; Secondary: 11N37.The first author was supported by ANID/Fondecyt/1200897 and Grant Basal-ANID AFB170001. a r X i v : . [ m a t h . D S ] J a n SEBASTIÁN DONOSO, ANH N. LE, JOEL MOREIRA, AND WENBO SUN preserving G -system . By a multiplicative measure preserving system , we mean a measurepreserving ( N , × ) -system with × being the multiplication on N .For x, y ∈ N , x + y is a perfect square if and only if x = k ( m − n ) and y = k (2 mn ) for k, m, n ∈ N and m > n . Therefore, to answer the case x + y of Conjecture 1.1 using theapproach in [11], it suffices to solve the following conjecture: Conjecture 1.2.
Let ( X, B , µ, T ) be a measure preserving ( N , × ) -system and A ∈ B with µ ( A ) > . There exist m, n ∈ N with m > n such that µ ( T − m − n A ∩ T − mn A ) > . The idea of using ergodic theoretical methods to solve combinatorial problems traces backto Furstenberg’s proof of Szemerédi’s Theorem [12] in 1977. The novelty in Frantzikinakisand Host [11]’s approach is the usage of additive averages to study multiplicative measurepreserving systems. This idea has the potential to address other unsolved problems in ergodicRamsey theory regarding partition regularity of polynomial equations and a primary goal ofthis paper is to begin a systematic study of this new tool and to obtain potential applications.1.1.
Additive averages of multiplicative recurrence sequences.
Throughout this pa-per, for a finite set E and function f : E → C , we use E n ∈ E f ( n ) to denote the average | E | (cid:80) n ∈ E f ( n ) . By the von Neumann ergodic theorem, for a multiplicative measure preserv-ing system ( X, B , µ, T ) , a set A ∈ B with µ ( A ) > and a multiplicative Følner sequence (Φ N ) N ∈ N in N (see Section 2 for definitions), we have(1) lim N →∞ E n ∈ Φ N µ ( A ∩ T − n A ) > . On the other hand, it is not clear whether (1) is still true if (Φ N ) N ∈ N is replaced with theadditive Følner sequence ([ N ]) N ∈ N . (Here [ N ] denotes the set { , . . . , N } .)Our first result confirms that this is indeed the case. In fact, we obtain a more generaltheorem regarding multiple ergodic averages along subsemigroups of ( N , × ) . Theorem 1.3.
Let ( X, B , µ, T ) be a measure preserving ( N , × ) -system and A ∈ B with µ ( A ) > . Let G be a subsemigroup of ( N , × ) . Then for every (cid:96) ∈ N , lim sup N →∞ E n ∈ G ∩ [ N ] µ ( A ∩ T − n A ∩ . . . ∩ T − n (cid:96) A ) > . Theorem 1.3 follows from the slightly more general Theorem 5.6 below. Besides N itself,examples of multiplicative semigroups of N include { m + Dn : m, n ∈ N } for some D ∈ N , { a n b m : m, n ∈ N } and { m a n b q c : m, n, q ∈ N } for a, b, c ∈ N . We remark that there are twonatural ways to take averages along these semigroups. One way is to enumerate its elementsin increasing order as done in Theorem 1.3. Alternatively, one can exploit the fact that these DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 3 semigroups admit parametrizations and take averages on the parameters. Our next resultstates that, also with this scheme, the averages of multiple recurrence sequences are positive.To formulate our result we need the notion of parametrized multiplicative function , whichdescribes certain parametrizations of multiplicative semigroups. We postpone the formaldefinition to Section 5.4, but here are some illustrative examples of commutative parametrizedmultiplicative functions: • f : N → N defined by f ( n , n ) = n + Dn , for any D ∈ N , • f : N → N defined by f ( n , n ) = | n + n n − n | , • f : N → N defined by f ( n , n , n ) = n a n a n a for any a , a , a ∈ N . Theorem 1.4.
Let ( X, B , µ, T ) be a measure preserving ( N , × ) -system and A ∈ B with µ ( A ) > . Let f : N k → N be a commutative parametrized multiplicative function and (cid:96) ∈ N . Then lim inf N →∞ E n ∈ [ N ] k µ (cid:16) A ∩ T − f ( n ) A ∩ T − f ( n ) A ∩ · · · ∩ T − f ( n ) (cid:96) A (cid:17) > . Theorem 1.4 follows from the more general Theorem 5.18 below. As an illustrative specialcase of Theorem 1.4 we obtain
Proposition 1.5.
Let ( X, B , µ, T ) be a measure preserving ( N , × ) -system and let A ∈ B with µ ( A ) > . Then lim inf N →∞ E m,n ∈ [ N ] µ ( A ∩ T − m + n A ) > . Topological recurrence and applications to number theory.
A subset R of theset of positive rational numbers Q > is called a set of (measurable) multiplicative recurrence iffor every ( Q > , × ) -system ( X, B , µ, T ) and A ∈ B with µ ( A ) > , there exists r ∈ R for which µ ( A ∩ T − r A ) > . By Furstenberg’s correspondence principle, R is a set of multiplicativerecurrence if and only if for any E ⊂ N of positive upper multiplicative density (see Section 2for definitions), there exist x, y ∈ E such that x/y ∈ R . If Conjecture 1.2 is true, then itfollows that the set { ( m − n ) / (2 mn ) : m, n ∈ N , m > n } is a set of multiplicative recurrence.There is a weaker notion that suffices for our purpose of studying partition regularity calledsets of topological multiplicative recurrence. A set R ⊂ Q > is called a set of topologicalmultiplicative recurrence if for every minimal topological ( Q > , × ) -system ( X, T ) and U ⊂ X open and non-empty, there exists r ∈ R such that U ∩ T − r U (cid:54) = ∅ . It turns out that R is a setof topological multiplicative recurrence if and only if for any finite coloring of N , there exist x, y of the same color such that x/y ∈ R (see Proposition 2.11 below).In Section 2 we show that if φ : ( N , +) → ( Q > , × ) is a homomorphism and S ⊂ N isset of topological (or measurable) additive recurrence, then φ ( S ) is a set of topological (ormeasurable) multiplicative recurrence. Using this fact, and drawing on the many knownexamples of sets of additive recurrence (cf. [12, 16, 20]), one can exhibit examples of sets of SEBASTIÁN DONOSO, ANH N. LE, JOEL MOREIRA, AND WENBO SUN multiplicative recurrence: { P ( n ) : n ∈ N } where P ∈ Z [ x ] satisfies P (0) = 0 and { p − : p prime } . However, all such examples are confined to a sparse semigroup of the form { a n : n ∈ N } , and there are hardly any other known examples of sets of multiplicative recurrence.Our next result produces a set of multiplicative recurrence which does not belong to theaforementioned class. Theorem 1.6.
Let a, b, c, d, (cid:96) ∈ Z , a, c, (cid:96) > and let R := (cid:26)(cid:16) an + bcn + d (cid:17) (cid:96) : n ∈ N (cid:27) .(1) If a (cid:54) = c , then R is not a set of topological multiplicative recurrence.(2) If a = c and either a | b or a | d , then R is a set of topological multiplicative recurrence. Theorem 1.6 follows from combining Propositions 3.2 and 3.5.A function f : N → C is called completely multiplicative if f ( mn ) = f ( m ) f ( n ) for m, n ∈ N .Klurman and Mangerel [17] proved that for any completely multiplicative function f with | f ( n ) | = 1 for n ∈ N , one has lim inf n →∞ | f ( n + 1) − f ( n ) | = 0 . They remarked that theirproof can be modified to treat the case in which the shift is replaced by any fixed k ∈ N .As an application of Theorem 1.6, we give a strengthening of their result by allowing botharbitrary shifts and dilations. Corollary 1.7.
Let f : N → C be a completely multiplicative function with | f ( n ) | = 1 for n ∈ N . Then for all a, k ∈ N , lim inf n →∞ | f ( an + k ) − f ( an ) | = 0 . The proof of Corollary 1.7 is given in Section 3, after the proof of Proposition 3.5.In the previous corollary, by setting f = ξ Ω( n ) , where ξ is a q -th root of and, for n ∈ N , Ω( n ) is the number of prime factors of n counting with multiplicity, one immediately derivesthe following enhancement of [17, Corollary 1.3]. Corollary 1.8.
For any a, q, k ∈ N , there exists infinitely many n ∈ N such that Ω( an + k ) ≡ Ω( an ) mod q . Using (an appropriate version of) the Poincaré recurrence theorem it can be shown thatevery subsemigroup of ( Q > , × ) is a set of multiplicative recurrence. Therefore, to understandwhich algebraic sets are sets of multiplicative recurrence, one needs to know which contain asubsemigroup. Our next theorem characterizes all polynomials of integer coefficients whoseimages contain subsemigroups of ( Q > , × ) . Theorem 1.9.
For P ∈ Z [ x ] , the set { P ( n ) : n ∈ N } contains an infinite subsemigroup of ( Q > , × ) if and only if P ( x ) = ( ax + b ) d for some a, d ∈ N , b ∈ Z with a | b ( b − . Theorem 1.9 follows from the more general Theorem 4.3.
DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 5
Regarding linear polynomials, we show that containing subsemigroups is not only a suffi-cient but also a necessary condition for their images to be sets of multiplicative recurrence.
Theorem 1.10.
For a, b ∈ N , the set { an + b : n ∈ N } is a set of multiplicative recurrence ifand only if a | b ( b − . Theorem 1.10 is implied by Theorem 3.9 below.Lastly, as previously mentioned, it is unknown whether the sets arising from Pythagoreantriples are sets of multiplicative recurrence. However, we show that these sets do not containsubsemigroups of ( Q > , × ) . Proposition 1.11.
Neither of the sets (cid:26) m + n mn : m, n ∈ N (cid:27) and (cid:26) mnm − n : m, n ∈ N , m > n (cid:27) contain an infinite subsemigroup of ( Q > , × ) . Proposition 1.11 is proved in Section 4, after Lemma 4.4.
Organization of the paper.
In Section 2 we give the background regarding sets of mul-tiplicative recurrence in both topological and measurable settings. Section 3 is devoted tostudying the topological recurrence properties of certain algebraic sets and providing ap-plications in number theory. In particular, Theorem 1.6, Corollary 1.7, Corollary 1.8 andTheorem 1.10 are proved in this section. In Section 4, we analyze polynomial configurationsand investigate when the image of N under a polynomial map contains a semigroup of ( N , × ) .Section 5 contains the ergodic theoretical point of view of this article where Theorem 1.3and Theorem 1.4 will be proved. Finally, in Section 6, we study the problem of findingPythagorean pairs and triples in dense subsets of finite fields. Acknowledgements.
We thank Florian Richter for help with the proof of Proposition 3.4and thank Vitaly Bergelson for helpful conversations.2.
Sets of multiplicative recurrence
In this section we collect basic facts about sets of recurrence for the semigroup ( N , × ) of natural numbers and the semigroup ( Q > , × ) of positive rational numbers, both undermultiplication. Most of the content is well known and presented here for completeness. Notations.
In this article, N = { , . . . . } denotes the set of positive natural numbers, and Q > denotes the set of positive rational numbers. For a positive natural number N , we let [ N ] denote the set { , . . . , N } . SEBASTIÁN DONOSO, ANH N. LE, JOEL MOREIRA, AND WENBO SUN
Multiplicatively invariant density and Furstenberg’s correspondence. A Følnersequence in a countable semigroup G is a sequence Φ = (Φ N ) N ∈ N of finite subsets of G whichare asymptotically invariant in the sense that for any g ∈ G , lim N →∞ | g Φ N ∩ Φ N || Φ N | = 1 , where g Φ N = { gx : x ∈ Φ N } . Given a Følner sequence Φ on a semigroup G , the upper density of a set A ⊂ G with respect to Φ is the quantity ¯ d Φ ( A ) := lim sup N →∞ | A ∩ Φ N || Φ N | . The upper Banach density of A is d ∗ ( A ) := sup Φ ¯ d Φ ( A ) , where the supremum is taken overall Følner sequences Φ on G . This supremum is always achieved, so for each set A there is aFølner sequence Φ such that d ∗ ( A ) = ¯ d Φ ( A ) .A discussion of Følner sequences and invariant densities in the case where G = ( N , × ) isprovided in [4]. As a concrete example one can take the sequence Φ N := { x ∈ N : x | N ! } , butthe exact choice will not be important in our discussion. Note that any Følner sequence for ( N , × ) is also a Følner sequence for ( Q > , × ) .Given a semigroup G , a measure preserving G -system is a measure preserving system ( X, B , µ, T ) where ( X, B , µ ) is a probability space and T = ( T g ) g ∈ G is an action of G on X by measure preserving transformations T g : X → X which satisfy T gh = T g ◦ T h for every g, h ∈ G .The Furstenberg correspondence principle connects sets with positive upper density andmeasure preserving systems. Here is the version we will use. Given a subset E of a semigroup G and n ∈ G we let E/n denote { m ∈ G : mn ∈ E } . Theorem 2.1 (Furstenberg’s correspondence principle for semigroups, cf. [5, Theorem 5.8and Remark 5.9]) . Let G be a countable commutative semigroup, let E ⊂ N and let Φ bea Følner sequence in G . Then there exists a G -system ( X, B , µ, T ) and a set A ∈ B with µ ( A ) = ¯ d Φ ( E ) and, for every n , . . . , n k ∈ G , ¯ d Φ (cid:0) E ∩ ( E/n ) ∩ ( E/n ) ∩ · · · ∩ ( E/n k ) (cid:1) ≥ µ (cid:0) A ∩ T − n A ∩ · · · ∩ T − n k A (cid:1) . Sets of multiplicative recurrence.
Sets of recurrence for ( Z , +) -systems, and theirconnection to combinatorics were introduced by Furstenberg in his book [13]. Sets of recur-rence for arbitrary countable semigroups were defined and studied in [7], and the specific casewhen the semigroup is ( N , × ) in [4]. Definition 2.2 (Cf. [7, Definition 1.1]) . Let G be a semigroup and let R ⊂ G . We say that R is a set of recurrence , or a set of measurable recurrence if for any measure preserving action DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 7 T = ( T g ) g ∈ G of G on a probability space ( X, µ ) and any A ⊂ X with µ ( A ) > there exists r ∈ R such that µ ( A ∩ T − r A ) > .Examples of sets of recurrence include G itself and any subsemigroup of G . In particular,note that if G admits an identity e , then the singleton { e } is a set of recurrence. Moregenerally, the usual proof of Poincaré’s recurrence theorem can be adapted to show that any ∆ -set (i.e., a set of the form { g i g i +1 · · · g j : i ≤ j } for some sequence ( g i ) i ∈ N in G ) is a set ofrecurrence. Proposition 2.3.
Let
G, H be semigroups and let φ : G → H be a homomorphism. If R ⊂ G is a set of recurrence, then φ ( R ) is a set of recurrence in H .Proof. Composing any measure preserving action of H with φ yields a measure preservingaction of G on the same space, and the conclusion follows. (cid:3) We are mainly interested in the semigroup ( N , × ) . In view of the following lemma, we willalso consider the group ( Q > , × ) . We denote by d ∗× ( A ) the Banach upper density of A withrespect to the multiplicative structure in either ( N , × ) or ( Q > , × ) . Lemma 2.4.
Let R ⊂ Q > . Then the followings are equivalent:(1) R is a set of recurrence, i.e., for any measure preserving action T of ( Q > , × ) on aprobability space ( X, µ ) and any A ⊂ X with µ ( A ) > there exists r ∈ R such that µ ( A ∩ T − r A ) > .(2) For any measure preserving action T of ( N , × ) on a probability space ( X, µ ) and any A ⊂ X with µ ( A ) > there exists a, b ∈ N with a/b ∈ R such that µ ( T − a A ∩ T − b A ) > .(3) For any A ⊂ Q > with d ∗× ( A ) > there exists r ∈ R such that A ∩ A/r (cid:54) = ∅ .(4) For any A ⊂ Q > with d ∗× ( A ) > there exists r ∈ R such that d ∗× ( A ∩ A/r ) > .(5) For any A ⊂ N with d ∗× ( A ) > there exist a, b ∈ N with a/b ∈ R such that A/a ∩ A/b (cid:54) = ∅ .(6) For any A ⊂ N with d ∗× ( A ) > there exists= a, b ∈ N with a/b ∈ R such that d ∗× ( A/a ∩ A/b ) > .Proof. The implications (6) ⇒ (5) and (4) ⇒ (3) are trivial. The implications (1) ⇒ (4) and(2) ⇒ (6) follow directly from the Furstenberg correspondence principle, Theorem 2.1.The implications (3) ⇒ (1) and (5) ⇒ (2) have very similar proofs, based on an observationof Bergelson, so here we only give the proof of (5) ⇒ (2). Suppose we are given a measurepreserving action T of ( N , × ) on a probability space ( X, µ ) and a set A ⊂ X with µ ( A ) > .Applying [4, Theorem 3.14] we find a set P ⊂ N with d ∗× ( P ) > and such that for any a, b ∈ P , µ ( T − a A ∩ T − b A ) > . Using (5) we can then find a, b ∈ P with a/b ∈ R , establishing (2). SEBASTIÁN DONOSO, ANH N. LE, JOEL MOREIRA, AND WENBO SUN
The implication (2) ⇒ (1) follows from the fact that any measure preserving action of ( Q > , × ) induces a natural measure preserving action of the subsemigroup ( N , × ) . Finally,the implication (1) ⇒ (2) follows from a routine inverse extension argument, which we providefor completeness.Let ( T n ) n ∈ N be a measure preserving action of ( N , × ) on ( X, µ ) . The invertible extensionof the measure preserving system ( X, µ, ( T n ) n ∈ N ) is the system ( Y, ν, ( S u ) u ∈ Q > ) where Y = (cid:110) ( x u ) u ∈ Q > ∈ X Q > : ( ∀ n ∈ N )( ∀ u ∈ Q > ) x un = T n x u (cid:111) ,S v ( x u ) u ∈ Q > = ( x uv ) u ∈ Q > ⊂ X Q > and for any cylinder set C = (cid:81) u ∈ F A u × (cid:81) u ∈ Q > \ F X where F ⊂ Q > is finite and A u ⊂ X are arbitrary measurable sets we have ν ( C ) = µ ( (cid:84) u ∈ F T − un A u ) , where n ∈ N is any common denominator of all elements of F . It is routineto check that ( Y, ν, ( S u ) u ∈ Q > ) is indeed a measure preserving system, and that the projectiondefined by ( x u ) u ∈ Q > (cid:55)→ x is a factor map between the ( N , × ) systems ( Y, ν, ( S u ) u ∈ N ) and ( X, µ, ( T n ) n ∈ N ) . On the other hand, the fact that (1) holds for the system ( Y, ν, ( S u ) u ∈ Q > ) implies that (2) holds for the system ( Y, ν, ( S u ) u ∈ N ) . But the property (2) passes through tofactors, so it also holds for the original system ( X, µ, ( T n ) n ∈ N ) . (cid:3) A weaker notion of recurrence is topological recurrence . Given a semigroup G , a topological G -system is a pair ( X, T ) where X is a compact metric space and T = ( T g ) g ∈ G is an action of G on X via continuous functions. Recall that a G -system is minimal if every orbit orb( x, T ) := { T g x : g ∈ G } is dense in X . Definition 2.5.
Let G be a semigroup and let R ⊂ G . We say that R is a set of topologicalrecurrence if for any minimal G -system and any non-empty open set U ⊂ X there exists r ∈ R such that U ∩ T − r U (cid:54) = ∅ .One can not drop the minimality condition entirely in this definition, as otherwise no setwould satisfy it. However, it is possible to slightly relax the minimality condition and stillhave an equivalent definition. Recall that given a topological G -system ( X, T ) , a point x ∈ X is called a minimal point if its orbit closure orb( x, T ) is minimal. Proposition 2.6.
Let G be a semigroup and let R ⊂ G . Then R is a set of topologicalrecurrence if and only if for any topological G -system ( X, T ) with a dense set of minimalpoints, and any non-empty open set U ⊂ X there exists r ∈ R such that U ∩ T − r U (cid:54) = ∅ .Proof. Let x ∈ U be a minimal point, let Y = orb( x, T ) and let V := Y ∩ U (cid:54) = ∅ . Then ( Y, T ) is a minimal system and V a non-empty open subset of Y , so there exists r ∈ R such that V ∩ T − r V (cid:54) = ∅ , which implies that U ∩ T − r U (cid:54) = ∅ . (cid:3) DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 9 If G is amenable (in particular if G is commutative) then in view of Bogolyubov-Krilov’stheorem [8], every set of recurrence is a set of topological recurrence. We remark in passingthat this is not true in general for non-amenable semigroups, and in fact Bergelson conjecturesthat it is never true for non-amenable groups (cf. [6] or [3, Conjecture after Exercise 20]).One can also ask whether every set of topological recurrence is a set of measurable recur-rence. When G = ( N , +) , a negative answer was given by Kriz [18]. We show below that wecan use Kriz’s example to answer the question in the negative for the semigroup ( N , × ) . Proposition 2.7.
Let
G, H be commutative semigroups and let φ : G → H be a homomor-phism. If R ⊂ G is a set of topological recurrence, then φ ( R ) is a set of topological recurrencein H .Proof. Let T be a minimal continuous action of H on the compact metric space X and let U ⊂ X be open and non-empty. Let S be the G -action on X defined by S g = T φ ( g ) , let x ∈ U ,let Y := { S g x : g ∈ G } and let Y ⊂ Y be a minimal S -subsystem. Since T is minimal, thereexists h ∈ H such that V := T − h U ∩ Y (cid:54) = ∅ .The set V is a non-empty open subset of Y , so by the hypothesis there exists g ∈ R suchthat S − g V ∩ V (cid:54) = ∅ . This implies that S − g T − h U ∩ T − h U (cid:54) = ∅ , and hence T − φ ( g ) U ∩ U (cid:54) = ∅ aswell. (cid:3) Corollary 2.8.
There exists a set R ⊂ N which is a set of topological recurrence in ( N , × ) but not a set of measurable recurrence in ( N , × ) .Proof. Let R ⊂ N be a set of additive topological recurrence which is not a set of additiverecurrence. Let φ : N → N be the map φ ( n ) = 2 n . Observe that φ : ( N , +) → ( N , × ) is ahomomorphism, so in view of Proposition 2.7 the set φ ( R ) is a set of multiplicative topologicalrecurrence.Let ( X, µ, T ) be an additive measure preserving system, and A ⊂ X satisfy µ ( A ) > but µ ( A ∩ T − n A ) = 0 for all n ∈ R . Consider the multiplicative measure preserving action S on ( X, µ ) given by S n = T a ( n ) , where a ( n ) is the unique integer such that n = 2 a ( n ) b ( n ) , with b ( n ) odd. Since a ( φ ( n )) = n for all n ∈ N , it follows that µ ( S − m A ∩ A ) = 0 for all m ∈ φ ( R ) ,and hence that φ ( R ) is not a set of multiplicative recurrence. (cid:3) Remark . We note that in general, shifts and dilations of sets of measurable multiplica-tive recurrence are not sets of measurable or even topological multiplicative recurrence. Forexample, the set { n : n ∈ N } is a set of multiplicative recurrence since it is a semigroup.On the other hand, { n + 3 : n ∈ N } is not since it is contained in N + 3 and it is shownin Theorem 1.10 that the latter set is not a set of topological multiplicative recurrence. Fordilations, the set { · n : n ∈ N } is not a set of multiplicative recurrence as seen with the following coloring: For n ∈ N , write n = 3 k q where k ∈ N ∪ { } and q ∈ N with (cid:45) q . Thencolor c ( n ) = 0 if k is even and c ( n ) = 1 if k is odd.The following is a direct corollary of Propositions 2.3 and 2.7. Corollary 2.10.
Let R ⊂ Q > be a set of multiplicative (topological) recurrence and let (cid:96) ∈ Z .Then the set { r (cid:96) : r ∈ R } is also a set of multiplicative (topological) recurrence. Proposition 2.11.
Let R ⊂ Q > \ { } . Then the followings are equivalent:(1) R is a set of topological recurrence, i.e. for any minimal action T of ( Q > , × ) on acompact metric space X and any non-empty open set U ⊂ X there exists r ∈ R suchthat U ∩ T − r U (cid:54) = ∅ .(2) For any topological ( Q > , × ) -system ( X, T ) with a dense set of minimal points and anynon-empty open set U ⊂ X there exist infinitely many r ∈ R such that U ∩ T − r U (cid:54) = ∅ .(3) For any minimal action T of ( N > , × ) on a compact metric space X and any non-empty open set U ⊂ X there exist infinitely many pairs ( a, b ) ∈ N with a/b ∈ R suchthat T − a U ∩ T − b U (cid:54) = ∅ .(4) For any finite partition Q > = C ∪ · · · ∪ C s there exists i ∈ { , . . . , s } , r ∈ R suchthat C i ∩ C i /r (cid:54) = ∅ .(5) For any finite partition N = C ∪ · · · ∪ C s there exist i ∈ { , . . . , s } , a, b ∈ N with a/b ∈ R such that C i ∩ C i /r (cid:54) = ∅ .Moreover, any of the conditions from Lemma 2.4 implies any of the conditions from thislemma.Proof. The proof is also routine, with the exception of the implication (1) ⇒ (3). Given aminimal multiplicative topological system ( X, ( T n ) n ∈ N ) , let ( Y, ( S u ) u ∈ Q > ) be its invertibleextension, defined so that Y = (cid:110) ( x u ) u ∈ Q > ∈ X Q > : ( ∀ n ∈ N )( ∀ u ∈ Q > ) x un = T n x u (cid:111) ⊂ X Q > and S v ( x u ) u ∈ Q > = ( x uv ) u ∈ Q > . Given a cylinder set C = (cid:81) u ∈ F A u × (cid:81) u ∈ Q > \ F X ⊂ X Q > ,where F ⊂ Q > is finite and A u ⊂ X are open sets, we have C ∩ Y (cid:54) = ∅ if and only if (cid:84) u ∈ F T − un A u (cid:54) = ∅ , where n ∈ N is any common denominator of all elements of F . For anysuch C and any x ∈ Y , since ( X, T ) is minimal the orbit of x n − is dense in X under T andhence for some m ∈ N we have T m x n − ∈ (cid:84) u ∈ F T − un A u so that T nmu x n − ∈ A u for all u ∈ F .Since x ∈ Y and nmu ∈ N , we have that T nmu x n − = x mu , and hence ( S m x ) u = x mu ∈ A u for all u ∈ F . We conclude that ( Y, ( S m ) m ∈ N ) is also a minimal ( N , × ) system. Since theprojection defined by ( x u ) u ∈ Q > (cid:55)→ x is a factor map between ( Y, ( S m ) m ∈ N ) and ( X, T ) ,the property (3) passes through to factors, and the property (1) for ( Y, ( S u ) u ∈ Q > ) impliesproperty (3) for ( Y, ( S m ) m ∈ N ) , this finishes the proof. (cid:3) DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 11
We remark in passing that items (4) and (6) of Lemma 2.4 also have an analogue fortopological recurrence, using the notion of piecewise syndetic set.3.
Sets of multiplicative recurrence and applications
Rational functions.
The purpose of this section is to prove Theorem 1.6. We start withthe following necessary criteria for a set to be a set of topological multiplicative recurrence.
Lemma 3.1.
For any (cid:15), M ∈ R satisfying < (cid:15) < < (cid:15) < M , the set S :=(( (cid:15), − (cid:15) ) ∪ (1 + (cid:15), M )) ∩ Q is not a set of topological multiplicative recurrence. In otherwords, if S does not have , or ∞ as an accumulation point, then S is not a set of topologi-cal multiplicative recurrence.Proof. Let (cid:15) and M be as in the hypothesis of the lemma. Choose c , c , a ∈ R such that c < − (cid:15) < (cid:15) < c and a > max { c /(cid:15), M/c } . Then it follows that (cid:91) k ∈ Z ( c a k , c a k ) ⊂ (0 , (cid:15) ) ∪ (1 − (cid:15), (cid:15) ) ∪ ( M, ∞ ) . Let S = { z ∈ C : | z | = 1 } and consider topological ( Q > , × ) -system ( S , T ) defined as T q ( e ( x )) = e (log a q ) e ( x ) for every q ∈ Q > and x ∈ [0 , (here e ( x ) := e πix ). Let (cid:15) = − log a c , (cid:15) = log a c and let B = { e ( x ) : x ∈ (1 − (cid:15) , (cid:15) ) } be a neighborhood of e (0) in S . We have T q e (0) ∈ B if any only if log a q ∈ (cid:83) k ∈ Z ( k − (cid:15) , k + (cid:15) ) , or equivalently q ∈ (cid:83) k ∈ Z ( a − (cid:15) a k , a (cid:15) a k ) . It follows that { q ∈ Q > : T q e (0) ∈ B } ⊂ (cid:91) k ∈ Z ( a − (cid:15) a k , a (cid:15) a k ) ⊂ (0 , (cid:15) ) ∪ (1 − (cid:15), (cid:15) ) ∪ ( M, ∞ ) . Since every set of topological multiplicative recurrence must intersect { q ∈ Q > : T q e (0) ∈ B } we conclude that R must intersect (0 , (cid:15) ) ∪ (1 − (cid:15), (cid:15) ) ∪ ( M, ∞ ) . Therefore the set S definedin Lemma 3.1 is not a set of topological multiplicative recurrence. (cid:3) We now prove part (1) of Theorem 1.6.
Proposition 3.2 (Part (1) of Theorem 1.6) . Let a, b, c, d, (cid:96) ∈ N with a (cid:54) = c . Then the set S = { ( an + b ) (cid:96) / ( cn + d ) (cid:96) : n ∈ N } is not a set of topological multiplicative recurrence.Proof. First note that ( an + b ) (cid:96) / ( cn + d ) (cid:96) → ( a/c ) (cid:96) as n → ∞ . Therefore, for any (cid:15) > ,the set S \ (( a/c ) (cid:96) − (cid:15), ( a/c ) (cid:96) + (cid:15) ) is finite. Moreover, since a (cid:54) = c , for sufficiently small (cid:15) , wehave (( a/c ) (cid:96) − (cid:15), ( a/c ) (cid:96) + (cid:15) ) does not contain , . Hence, S does not have , or ∞ as an accumulation point, and by Lemma 3.1, the set S is not a set of topological multiplicativerecurrence. (cid:3) For the proof of part (2) of Theorem 1.6, we need a recurrence result in topological dynam-ical systems:
Lemma 3.3.
Let ( X, T ) be a topological ( N , +) -system where X is a compact metric spacewith the metric d . Suppose that there exists x ∈ X such that { T n x : n ∈ N } is dense in X .Then for every y ∈ X, (cid:15) > and N ∈ N , there exists m ∈ N such that d ( T i + m x, T i y ) < (cid:15) for all i ∈ [ N ] .Proof. For i ∈ [ N ] , the map T i : X → X is continuous, hence uniformly continuous. Thus,there exists δ > such that for w, z ∈ X , d ( w, z ) < δ implies d ( T i w, T i z ) < (cid:15) for all i ∈ [ N ] . Since { T n x : n ∈ N } is dense in X , for any y ∈ Y we can find m ∈ N such that d ( T m x, y ) < δ . Then the choice of δ implies that d ( T i + m x, T i y ) < (cid:15) for all i ∈ [ N ] . (cid:3) Next we prove the following special case of part (2) of Theorem 1.6.
Proposition 3.4.
For any a ∈ N , the sets { ( an + 1) /an : n ∈ N } and { ( an − / ( an ) : n ∈ N } are sets of topological multiplicative recurrence.Proof. We only prove the first set is a set of topological multiplicative recurrence as the prooffor the second set is the same. Fix r ∈ N with r ≥ . Let N denote N ∪ { } and let X = [ r ] N = { ( w ( n )) n ∈ N : w ( n ) ∈ [ r ] } . Define a metric d on X as the following: For w = ( w ( n )) n ∈ N and z = ( z ( n )) n ∈ N , d ( w, z ) : = (cid:88) n ∈ N | w ( n ) − z ( n ) | n . Under this metric, X becomes a compact metric space and d ( w, z ) < implies w (0) = z (0) .Let T : X → X be the left shift which is defined as T ( w ( n )) n ∈ N = ( w ( n + 1)) n ∈ N . Then T is a continuous map.Let c : N → [ r ] be a finite coloring of N . We need to find x, y ∈ N of the same color and n ∈ N such that y/x = ( an + 1) /an . This is equivalent to finding { x, x + k } of the same colorwith ak | x (indeed, writing x = nak and y = x + k we have y/x = ( an + 1) /an ). By regarding c as an element of X , we define X to be the closure of { T n c : n ∈ N } in X . It follows that ( X, T ) is a topological ( N , +) -system. Let ( Y, T ) be a minimal subsystem of ( X, T ) . Foreach color j ∈ [ r ] , let U j be the cylinder U j = { x ∈ X : x (0) = j } . By minimality, the returntimes of points in Y ∩ U j back to Y ∩ U j are uniformly bounded, i.e. there exists N j ∈ N suchthat for all y ∈ Y ∩ U j , there exists n = n ( y ) ∈ [ N j ] satisfying T n y ∈ U j ∩ Y (for example, The existence of minimal subsystems follows from an application of Zorn’s lemma (cf. [1, Theorem 1.4]).
DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 13 see [1, Chapter 1]). Setting N = max { N j : j ∈ [ r ] } , we have that for all y ∈ Y and n ∈ N ,there exists ≤ n (cid:48) ≤ N for which y ( n ) = y ( n + n (cid:48) ) .Fix y ∈ Y . By Lemma 3.3, there exists m ∈ N such that d ( T i + m c, T i y ) < for all i ∈ [ aN !] (the number aN ! is chosen here because later we will use the fact that an (cid:48) | aN ! forall ≤ n (cid:48) ≤ N ). In particular, we have c ( m + i ) = y ( i ) for ≤ i ≤ aN ! . Let i ∈ [ aN !] satisfy m + i ≡ aN ! . From previous paragraph, there exists ≤ n (cid:48) ≤ N such that y ( i ) = y ( i + n (cid:48) ) . Hence it follows that c ( m + i ) = c ( m + i + n (cid:48) ) . By writing x = m + i and k = n (cid:48) , we have c ( x ) = c ( x + k ) with ak | x , finishing our proof. (cid:3) We are now ready to prove part (2) of Theorem 1.6.
Proposition 3.5 (Part (2) of Theorem 1.6) . For a, b, d, (cid:96) ∈ Z with a, (cid:96) > and a | b or a | d , S = (cid:40)(cid:18) an + ban + d (cid:19) (cid:96) : n ∈ N (cid:41) is a set of topological multiplicative recurrence.Proof. In view of Corollary 2.10, it suffices to show that S (cid:48) = (cid:26) an + ban + d : n ∈ N (cid:27) is a set of topological multiplicative recurrence. Without loss of generality, assume a | d and d = ad . If b = d , then S (cid:48) = { } which is trivially a set of multiplicative recurrence. If b > d ,then we have S (cid:48) = (cid:26) a ( n + d ) + b − ad a ( n + d ) : n ∈ N (cid:27) which contains the set (cid:26) a ( n + d ) + b − ad a ( n + d ) : n + d ≡ b − ad ) (cid:27) = (cid:26) am + 1 am : m ∈ N (cid:27) . By Proposition 3.4, the last set is a set of topological multiplicative recurrence, and hence sois S (cid:48) .On the other hand, if b < d , then S (cid:48) contains the set (cid:26) a ( n + d ) + b − ad a ( n + d ) : n + d ≡ ad − b ) (cid:27) = (cid:26) am − am : m ∈ N (cid:27) which is a again a set of topological multiplicative recurrence. (cid:3) Next we prove Corollary 1.7.
Proof of Corollary 1.7.
Let f : N → C be a completely multiplicative function with | f ( n ) | = 1 for n ∈ N and let a, k ∈ N . We need to prove that, lim inf n →∞ | f ( an + k ) − f ( an ) | = 0 . Let (cid:15) > . Consider the topological ( N , × ) -system ( S , T ) where S = { z ∈ C : | z | = 1 } and T n e πix = f ( n ) e πix for all x ∈ [0 , and n ∈ N . Let A = { e πix : x ∈ ( − (cid:15)/ , (cid:15)/ } ⊂ S . Then by Proposition 3.5 and Proposition 2.11, there exist infinitely many n ∈ N suchthat T ( an + k ) m A ∩ T anm A (cid:54) = ∅ for some m ∈ N (which may depend on n ). This implies f (( an + k ) m ) A ∩ f ( anm ) A (cid:54) = ∅ , or equivalently, f ( m ) f ( an + k ) A ∩ f ( m ) f ( an ) A (cid:54) = ∅ . Sincemultiplication by f ( m ) is an isometry, we have f ( an + k ) A ∩ f ( an ) A (cid:54) = ∅ . It follows that | f ( an + k ) − f ( an ) | ≤ (cid:15) for infinitely many n . Since (cid:15) is arbitrary, we have lim inf n →∞ | f ( an + k ) − f ( an ) | = 0 . (cid:3) Remark . We remark that the necessary condition described in Theorem 1.6 is not sufficientfor a set of the form S = { ( an + b ) (cid:96) / ( an + d ) (cid:96) : n ∈ N } to be a set of topological multiplicativerecurrence. Consider the set S = { (3 n + 1) / (3 n + 2) : n ∈ N } and the -Rado coloring : For m ∈ N , write m = 3 k q where k ∈ N ∪ { } and (cid:45) q . Then color c ( m ) = 1 if q ≡ and c ( m ) = 2 if q ≡ . It is easy to see that there do not exist m , m of the same colorsuch that m /m ∈ { (3 n + 1) / (3 n + 2) : n ∈ N } .Remark 3.6 naturally raises the following question, which asks whether the sufficient con-ditions identified in Theorem 1.6 for a set of the form S = { ( an + b ) / ( an + d ) : n ∈ N } to bea set of topological multiplicative recurrence are also necessary. Question 3.7.
For a, b, d, (cid:96) ∈ N , is it true that the set S = { ( an + b ) / ( an + d ) : n ∈ N } is aset of topological multiplicative recurrence if and only if a | b or a | d ? Remark . Theorem 1.6 implies that for any finite coloring of N and every k ∈ N , thereexist a, n ∈ N such that an | x and the set (cid:26) x, x an + kan (cid:27) is monochromatic. It is natural to ask if, more generally, we can find a, n ∈ N for which an | x and the set (cid:26) x, x an + 1 an , x an + 2 an , . . . , x an + kan (cid:27) is monochromaticIt turns out that the answer is negative, even in the case a = 1 and k = 2 , as shown by thefollowing example.Let c : N → { , } be the -Rado coloring as constructed in Remark 3.6. Note that inevery consecutive integers, there is a number congruent to and another congruentto , therefore there are no consecutive numbers with the same color. DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 15
On the other hand, observe that c ( x ) = c ( y ) if and only if c ( xz ) = c ( yz ) . Indeed, this isobvious if z is a power of ; and if z is co-prime to then c ( xz ) ≡ c ( x ) z mod 3 . If the set (cid:110) xn n, xn ( n + 1) , xn ( n + 2) (cid:111) were monochromatic for some x, n ∈ N with n | x , so would be the set { n, n + 1 , n + 2 } , acontradiction.3.2. Linear polynomials.
Let ( X, T ) be an ( N , × ) -system. If X is a finite set, we say that ( X, ( T n ) n ∈ N ) is a finite ( N , × ) -system. The following theorem classifies which sets of the form a N + b that are sets of multiplicative recurrence, and hence implies Theorem 1.10. Theorem 3.9.
Let a, b ∈ N and S = { an + b : n ∈ N } . The following are equivalent:(i) a | b ( b − .(ii) S is a multiplicative semigroup.(iii) S contains an infinite multiplicative semigroup.(iv) S is a set of measurable recurrence for ( N , × ) -systems.(v) S is a set of topological recurrence for finite ( N , × ) -systems.Proof. (i) ⇒ (ii): If a | b ( b − , then for any ax + b, ay + b ∈ S , we have ( ax + b )( ay + b ) = a (cid:18) axy + bx + by + b ( b − a (cid:19) + b ∈ S. (ii) ⇒ (iii): Obvious.(iii) ⇒ (iv): This follows from the Poincaré Recurrence Theorem.(iv) ⇒ (v): Obvious.(v) ⇒ (i): We prove the contrapositive. Let a, b be such that a (cid:45) b ( b − . It suffices toconstruct a finite ( N , × ) -system ( X, ( T n ) n ∈ N ) such that whenever n ∈ S , the permutation T n fixes no element. Let a (cid:48) = a/ ( a, b ) . Observe that b (cid:54)≡ a (cid:48) , for otherwise b ( b − wouldbe a multiple of a = ( a, b ) a (cid:48) . We need to consider separate cases:Case 1: ( a (cid:48) , b ) = 1 . In this case every n ∈ S satisfies ( n, a (cid:48) ) = 1 and n ≡ b (cid:54)≡ a (cid:48) .Let X = (cid:8) x ∈ { , , . . . , a (cid:48) − } : ( x, a (cid:48) ) = 1 (cid:9) . For n ∈ N , let T n : X → X be defined asfollows. Let A be the set of all q ∈ N such that the all prime divisors of q also divide a (cid:48) .Any natural number n can be written uniquely as n = uv = u ( n ) v ( n ) where u ∈ A, v ∈ N and ( v, a (cid:48) ) = 1 . For x ∈ X , define T n x = xv ( n ) mod a (cid:48) . Notice that v ( nm ) = v ( n ) v ( m ) ,so ( T n ) n ∈ N is indeed an action of ( N , × ) on X . Since every n ∈ S satisfies ( n, a (cid:48) ) = 1 and n (cid:54)≡ a (cid:48) , v ( n ) = n (cid:54)≡ a (cid:48) . Therefore, for n ∈ S , we have T n does not fix any pointin X .Case 2: ( a (cid:48) , b ) > . Let p be a prime which divides ( a (cid:48) , b ) and let k ∈ N be such that p k − | b but p k (cid:45) b . Let X = Z k and let T n : X → X , n ∈ N be defined as follows. For n ∈ N , let u ( n ) and v ( n ) be the unique non-negative integers satisfying n = p u ( n ) v ( n ) and p (cid:45) v ( n ) . For x ∈ X , define T n x := x + u ( n ) mod k . Since u ( nm ) = u ( n ) + u ( m ) , ( T n ) n ∈ N indeed definesan action of ( N , × ) on X . Because ( a (cid:48) , b/ ( a, b )) = 1 and p | a (cid:48) , it follows that p (cid:45) b/ ( a, b ) . Onthe other hand, p k − | b . Therefore, p k − | ( a, b ) , and hence p k | a = ( a, b ) a (cid:48) . Thus for any n ∈ S ,we have u ( n ) = k − , which implies that T n x = x − k for every x ∈ X . In particular, T n does not fix any element of X . (cid:3) A polynomial P ∈ Z [ x ] is called divisible if for any q ∈ N , there exists n ∈ N such that q | P ( n ) . As proven in [16], the set { P ( n ) : n ∈ N } is a set of additive topological (andmeasurable) recurrence if and only if P is divisible. Corollary 3.10.
Let P ∈ Z [ x ] . If S = { P ( n ) : n ∈ N } is a set of topological multiplicativerecurrence, then R = { P ( n )( P ( n ) −
1) : n ∈ N } is a set of measurable additive recurrence,i.e. the polynomial P ( P − is divisible.Proof. Assume P ( P − is not divisible. Then there exists a ∈ N such that a (cid:45) P ( n )( P ( n ) − for all n ∈ N . Hence a(2) { P ( n ) : n ∈ N } ⊂ (cid:91) ≤ b ≤ a − a (cid:45) b ( b − a N + b. In view of Theorem 3.9, for each b ∈ { , . . . , a − } such that a (cid:45) b ( b − , the set a N + b isnot a set of multiplicative recurrence. Invoking [7, Theorem 2.7 (b)] it follows that the unionin the right hand side of (2) can not be a set of recurrence. (cid:3) Example . As an application of Corollary 3.10, the set { n + 3 : n ∈ N } is not a set oftopological multiplicative recurrence. This is because the polynomial Q ( n ) = n + 5 n + 6 isnot divisible, as (cid:45) Q ( n ) for any n ∈ N .Any polynomial P ∈ Z [ x ] with P (0) = 0 is divisible, so in that case the set S = { P ( n ) : n ∈ N } is a set of measurable additive recurrence. However, this set is not a set of multiplicativerecurrence in general as the following example shows: Example . Let p be a prime. The set S = { pn : n ∈ N } is not a set of topologicalmultiplicative recurrence. Indeed, let E = { p k (cid:96) : k ∈ N ∪ { } , (cid:96) ∈ N , p (cid:45) (cid:96) } and F = { p k +1 (cid:96) : k ∈ N ∪ { } , (cid:96) ∈ N , p (cid:45) (cid:96) } . Then E ∪ F = N . For every x, y ∈ E or x, y ∈ F , x/y has the form p k (cid:96) /(cid:96) for some k ∈ Z and (cid:96) , (cid:96) ∈ N with p (cid:45) (cid:96) (cid:96) . It follows that x/y (cid:54)∈ S , hence S is nota set of multiplicative recurrence.Example 3.12 can also be used to show that the converse to Corollary 3.10 is false ingeneral. Indeed, take P ( n ) = pn . Then P ( n )( P ( n ) −
1) = pn ( pn − is divisible, but { P ( n ) : n ∈ N } is not set of topological multiplicative recurrence. DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 17 Algebraic sets that contain multiplicative semigroups
As mentioned in Section 2, every subsemigroup of ( N , × ) is a set of multiplicative recur-rence. With the ultimate goal of understanding which algebraically defined sets are sets ofmultiplicative recurrence, it is natural to also study sets which are (or contain) a semigroup.In this section we investigate this question for sets that are the image of a polynomial or theimage of a rational function.4.1. Polynomial sets that contain multiplicative semigroups.
In this section we proveTheorem 1.9. We first need some lemmas.
Lemma 4.1.
Let d ≥ and P ( x ) = a d x d + a d − x d − + . . . + a x + a ∈ Q [ x ] such that at leastone of a , a , . . . , a d − is non-zero and a d > . Then S = { P ( n ) : n ∈ N } does not containany infinite subsemigroup of ( Q > , × ) .Proof. First we show that, for any D ∈ N \ { } , there are only finitely many n ∈ N suchthat D d P ( n ) = P ( Dn ) . Indeed, otherwise, the polynomial D d P ( x ) − P ( Dx ) would haveinfinitely many roots, and hence D d P ( x ) = P ( Dx ) for all x ∈ R . It would follow that P ( D k ) = P (1)( D k ) d for all k ∈ N . This implies that the equation P ( x ) = P (1) x d hasinfinitely many solutions. So P ( x ) = P (1) x d for all x ∈ R . This is contradicts our assumptionthat at least one of a , a , . . . , a d − is nonzero.Next we assume, for the sake of a contradiction, that there exists D ∈ N \ { } such that { D n : n ∈ N } ⊂ S . Then there exist infinitely many m > n ∈ N such that(3) D d P ( n ) = P ( m ) . Using the first paragraph of the proof, there are infinitely many solutions to (3) with m (cid:54) = Dn .Next, observe that (3) can be written as(4) a d ( m d − ( Dn ) d ) = d − (cid:88) j =0 a j ( D d n j − m j ) . Note that for all x, y ∈ N with x (cid:54) = y , | x d − y d | > (max { x, y } ) d − . Hence if m (cid:54) = Dn forsome m, n ∈ N , then | m d − ( Dn ) d | > (max { m, Dn } ) d − . On the other hand, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d − (cid:88) j =0 a j ( D d n j − m j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ d −
1) max ≤ j ≤ d − {| a j |} (max { m, Dn } ) d − . Therefore, from (4) we derive that a d max { m, Dn } d − < d −
1) max ≤ j ≤ d − {| a j |} max { m, Dn } d − for infinitely many m, n ∈ N . This is impossible and hence it yields the desired contradiction. (cid:3) Lemma 4.2.
Let P ∈ Z [ x ] . Then there exist a, b ∈ Z and Q ∈ Q [ x ] such that P ( x ) = Q ( ax + b ) and writing Q ( x ) = a d x d + a d − x d − + · · · + a x + a we have a d − = 0 . If the leading coefficientof P is positive, then a > .Proof. Let P ( x ) = b d x d + · · · + b and factor it over C as b d (cid:81) di =1 ( x − α i ) , where α , . . . α d ∈ C are the roots of P counted with multiplicity. It follows that (cid:80) di =1 α i = − b d − /b d . Let a = db d , b = b d − and define Q ( y ) := P (( y − b ) /a ) . Observe that Q ( y ) = 0 if and only if y = aα i + b for some ≤ i ≤ d . Therefore the sum of the roots of Q is , which implies that a d − = 0 . (cid:3) We are ready to prove Theorem 1.9. In fact we prove the following more general result.
Theorem 4.3.
Let P ∈ Z [ x ] and S = { P ( n ) : n ∈ N } . The following are equivalent:(i) S is an infinite subsemigroup of ( Q > , × ) .(ii) S contains an infinite subsemigroup of ( Q > , × ) .(iii) P ( x ) = ( ax + b ) d for some a, d ∈ N , b ∈ Z with a | b ( b − .Proof. (iii) ⇒ (i): Assume Q ( x ) = ( ax + b ) d with a | b ( b − . Then for any x, y ∈ N , Q ( x ) Q ( y ) = ( ax + b ) d ( ay + b ) d = (cid:18) a (cid:18) axy + bx + by + b ( b − a (cid:19) + b (cid:19) d ∈ S. Hence S is a multiplicative semigroup.(i) ⇒ (ii): Trivial.(ii) ⇒ (iii): Assume that S = { P ( n ) : n ∈ N } contains an infinite multiplicative semigroupof Q > . Then the leading coefficient of P is positive. Write P ( x ) = Q ( ax + b ) as in Lemma 4.2,noting that a > . Then we have { P ( n ) : n ∈ N } = { Q ( an + b ) : n ∈ N } ⊂ { Q ( n ) : n ∈ N } .It then follows that { Q ( n ) : n ∈ N } contains a semigroup. By Lemma 4.1, Q ( x ) = a d x d forsome d ∈ N and a d ∈ Q . Since { Q ( n ) : n ∈ N } contains a semigroup, there exist x, y, z ∈ N such that Q ( x ) Q ( y ) = Q ( z ) . In other words, a d ( xy ) d = a d z d , or a d = ( z/ ( xy )) d . Let c = z/ ( xy ) ∈ Q > , we get a d = c d . Hence P ( x ) = ( c ( ax + b )) d = ( cax + cb ) d .By abuse of notation, assume that P ( x ) = ( ax + b ) d for some a, b ∈ Q . Since P ∈ Z [ x ] , wehave a, b ∈ Z . Because S contains an infinite multiplicative semigroup, there exists D ∈ N \{ } such that for all k ∈ N , there exists x ∈ N and ( ax + b ) d = D kd . In other words, ax + b = D k .It follows that D k ≡ b mod a for all k ∈ N . Hence we have simultaneously that D k ≡ b and D k ≡ b mod a . This implies that b ≡ b mod a , or a | b ( b − , which finishes the proof. (cid:3) Rational functions that do not contain a semigroup.
We prove Proposition 1.11in this section. In addition, we show that many other rational functions of interest do notcontain infinite multiplicative semigroups.
DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 19
Lemma 4.4.
Let (cid:96) ≥ , q , q , . . . , q (cid:96) ∈ R \{ } be non-zero and | q i | (cid:54) = | q j | for all i (cid:54) = j .Let a , . . . , a (cid:96) ∈ R be such that a q k + . . . + a (cid:96) q k(cid:96) = 0 for infinitely many k ∈ N . Then a = a = . . . = a (cid:96) = 0 .Proof. By contradiction, assume some a i are non-zero. By discarding those a i that are zero,without loss of generality, we may assume that none of the a i are zero.Assume | q | = max {| q i | : 1 ≤ i ≤ (cid:96) } > . Then we have | a q k | = | a q k + . . . + a (cid:96) q k(cid:96) | . It follows that | a | = | a ( q /q ) k + . . . + a (cid:96) ( q (cid:96) /q q ) k | for infinitely many k ∈ N . But this is impossible since the right hand side approaches zero as k → ∞ . (cid:3) We are ready to prove Proposition 1.11.
Proof of Proposition 1.11.
We only give a proof that the set S = { ( m + n ) / (2 mn ) : m, n ∈ N } does not contain an infinite subsemigroup of ( Q > , × ) as the proof for the other set isthe same. By contradiction, assume that there exist p, q ∈ Z , p (cid:54) = q, ( p, q ) = 1 such that { ( p/q ) k : k ∈ N } ⊂ S . By replacing ( p/q ) by ( p/q ) , we can assume p, q > . Then it followsthat for every k ∈ N , there exist m, n ∈ N such that(5) p k q k = m + n mn . We can dividing both m and n by gcd ( m, n ) and (5) still holds. Therefore, without loss ofgenerality, we may assume ( m, n ) = 1 . Then p k mn = q k ( m + n ) . Since ( p, q ) = 1 , it follows that q k | mn . On the other hand, since ( m, n ) = 1 , ( mn, m + n ) =1 . Thus mn | q k . It follows that q k = mn or q k = 2 mn .If q k = mn , then p k = m + n . Since ( m, n ) = 1 , m = q k and n = q k for some q q = q .Hence(6) p k = ( q ) k + ( q ) k . If q k = 2 mn , then p k = m + n . Similarly to above, m = ( q ) k / and n = q k for some q q = q . We then have p k = q k / q k , or equivalently,(7) p k = ( q ) k + 4( q ) k . Since the set { ( q , q ) ∈ N : q q = q } is finite, there must be some fixed q , q suchthat either (6) or (7) is true for infinitely many k ∈ N . But, in view of Lemma 4.4, this isimpossible, and hence we obtained the desired contradiction. (cid:3) Let (cid:96) , (cid:96) , (cid:96) , (cid:96) ∈ Z be pairwise distinct and let(8) S = (cid:26) ( m + (cid:96) n )( m + (cid:96) n )( m + (cid:96) n )( m + (cid:96) n ) : m, n ∈ N (cid:27) . It is shown in [11] that the set S in (8) is a set of measurable multiplicative recurrence. In thefollowing proposition, we show that S does not contain any infinite subsemigroup of ( Q > , × ) . Proposition 4.5.
Let (cid:96) , (cid:96) , (cid:96) , (cid:96) ∈ Z be pairwise distinct. Then the set S = (cid:26) ( m + (cid:96) n )( m + (cid:96) n )( m + (cid:96) n )( m + (cid:96) n ) : m, n ∈ N (cid:27) does not contain any infinite subsemigroup of ( Q > , × ) .Proof. By contradiction, assume there exist p ∈ Z , q ∈ N with ( p, q ) = 1 such that for all k ∈ N , there exist m, n ∈ N such that ( m + (cid:96) n )( m + (cid:96) n )( m + (cid:96) n )( m + (cid:96) n ) = p k q k . Dividing m, n by gcd ( m, n ) , we can assume ( m, n ) = 1 . We then have q k ( m + (cid:96) n )( m + (cid:96) n ) = p k ( m + (cid:96) n )( m + (cid:96) n ) . It follows that q k | p k ( m + (cid:96) n )( m + (cid:96) n ) . But since ( p, q ) = 1 , it implies that q k | ( m + (cid:96) n )( m + (cid:96) n ) . Similarly, p k | ( m + (cid:96) n )( m + (cid:96) n ) . Let t = ( m + (cid:96) n )( m + (cid:96) n ) p k = ( m + (cid:96) n )( m + (cid:96) n ) q k ∈ Z . Therefore t | gcd (( m + (cid:96) n )( m + (cid:96) n ) , ( m + (cid:96) n )( m + (cid:96) n )) .For i, j ∈ { , , , } , let d = gcd ( m + (cid:96) i n, m + (cid:96) j n ) . Then d | ( (cid:96) j − (cid:96) i ) n and d | (( (cid:96) j − (cid:96) i ) m +( (cid:96) j − (cid:96) i ) (cid:96) i n ) . Therefore, d | ( (cid:96) j − (cid:96) i ) m . Since ( m, n ) = 1 , it follows that d | ( (cid:96) j − (cid:96) i ) . We deducethat t | ( (cid:96) − (cid:96) )( (cid:96) − (cid:96) )( (cid:96) − (cid:96) )( (cid:96) − (cid:96) . We then have ( m + (cid:96) n )( m + (cid:96) n ) = tp k and ( m + (cid:96) n )( m + (cid:96) n ) = tq k . Since ( m + (cid:96) n, m + (cid:96) n ) | ( (cid:96) − (cid:96) ) , we deduce that there exists some k = k ( (cid:96) , (cid:96) , (cid:96) , (cid:96) ) suchthat(9) m + (cid:96) n = u p k − k and m + (cid:96) n = u p k − k DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 21 where ( p , p ) = 1 , p p = p and u u = tp k . Similarly, there exists k = k ( (cid:96) , (cid:96) , (cid:96) , (cid:96) ) such that(10) m + (cid:96) n = v q k − k and m + (cid:96) n = v q k − k where ( q , q ) = 1 , q q = q and v v = tq k .Since there are only finitely many choices of t, u i , v i , p i , q i for i = 1 , , it follows that thereexist fixed u i , v i , p i , q i for i = 1 , for which there are infinitely many k ∈ N such that (9) and(10) simultaneously have solutions ( m, n ) ∈ N . Solving these equations, we get n = u p k − k − u p k − k (cid:96) − (cid:96) = v q k − k − v q k − k (cid:96) − (cid:96) . In particular, u ( (cid:96) − (cid:96) ) p k × p k − u ( (cid:96) − (cid:96) ) p k × p k − v ( (cid:96) − (cid:96) ) q k × q k + v ( (cid:96) − (cid:96) ) q k × q k = 0 for infinitely many k ∈ N . Since p , p , q , q are pairwise coprime, we can apply Lemma 4.4to show that u i = v i = 0 for i = 1 , . It follows from (9) that m = n = 0 . This contradictsour assumption that m, n ∈ N finishing the proof. (cid:3) Additive averages of multiplicative recurrence sequences
Multiplicative subsemigroups of N . Given a subset S ⊂ G of a semigroup G and anelement g ∈ G , we write S/x := { y ∈ G : yx ∈ S } . We recall from Section 2 the notion ofupper density of S with respect to a Følner sequence Φ in G : ¯ d Φ ( S ) = lim sup N →∞ | S ∩ Φ N || Φ N | . The asymptotic invariance of Φ implies that that ¯ d Φ ( S/x ) = ¯ d Φ ( S ) for any x ∈ G .Let G be a countable discrete semigroup with a Følner sequence. A subset S ⊂ G is called (right) syndetic if there exists a finite set F ⊂ G such that S/F := (cid:83) x ∈ F S/x = G . Thefollowing proposition is well known and we include its proof for completeness. Proposition 5.1.
A set S is syndetic if and only if it has positive upper density with respectto any Følner sequence.Proof. If S is syndetic and Φ is a Følner sequence, then take F finite so that S/F = G . Wehave d Φ ( G ) ≤ (cid:80) x ∈ F ¯ d Φ ( S/x ) = ¯ d Φ ( S ) · | F | , so ¯ d Φ ( S ) > .Conversely, if S is not syndetic and Φ is a Følner sequence, for every N ∈ N we have that S/ Φ N (cid:54) = G . Take x N ∈ G \ ( S/ Φ N ) . Then Ψ N := x N Φ N is disjoint from S . On the otherhand, the sequence Ψ = (Ψ N ) N ∈ N is a Følner sequence, and so ¯ d Ψ ( S ) = 0 . (cid:3) The next result shows that in the case when ( G, × ) is a subsemigroup of ( N , × ) , thena multiplicative syndetic subset of G has positive “upper density” also with respect to the“additive” sequence G ∩ [ N ] . Lemma 5.2.
Let ( G, × ) be a subsemigroup of ( N , × ) and S be a syndetic subset of G . Then lim sup N →∞ E n ∈ G ∩ [ N ] S ( n ) > . Moreover, if G has positive lower additive density in N , then lim inf N →∞ E n ∈ G ∩ [ N ] S ( n ) > . Proof.
We first claim that for any non-empty set E ⊂ N and c > , we have(11) lim sup N →∞ | E ∩ [ N ] || E ∩ [ cN ] | > . In fact, by contraction, assume lim sup N →∞ | E ∩ [ N ] || E ∩ [ cN ] | = 0 . Then there exists N such that for all N ≥ N , we have | E ∩ [ N ] || E ∩ [ cN ] | < c . Or equivalently, | E ∩ [ cN ] | > c | E ∩ [ N ] | . Inductively, we have for all n ∈ N ,(12) | E ∩ [ c n N ] | > (2 c ) n | E ∩ [ N ] | . But since | E ∩ [ c n N ] | ≤ c n N , (12) cannot hold for n very large. This is a contradiction.This finishes the proof of the claim.We now return to the proof of Lemma 5.2. Since S is syndetic in G , there exists a finitesubset F ⊂ G such that S/F = G . It follows that (cid:88) m ∈ F S ( mn ) ≥ for all n ∈ G. Therefore(13) (cid:88) n ∈ [ N ] ∩ G (cid:88) m ∈ F S ( mn ) ≥ | G ∩ [ N ] | . Let M = max { m : m ∈ F } . Since each n ≤ M N is repeated at most | F | times in the sumappearing in the left hand side of (13), we deduce that (cid:88) n ∈ G ∩ [ MN ] S ( n ) ≥ | G ∩ [ N ] || F | . DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 23
Therefore(14) E n ∈ G ∩ [ MN ] S ( n ) ≥ | G ∩ [ N ] || F || G ∩ [ M N ] | In view of the claim,(15) lim sup N →∞ | G ∩ [ N ] || G ∩ [ M N ] | > . (15) and (14) imply d G ( S ) > .Assume G has positive lower additive density. In this case, we have | G ∩ [ N ] || G ∩ [ M N ] | = | G ∩ [ N ] | N · N | G ∩ [ M N ] | ≥ δM for sufficiently large N. Hence, for sufficiently large N ,(16) E n ∈ G ∩ [ MN ] S ( n ) ≥ δM | F | . We now have(17) E n ∈ G ∩ [ N ] S ( n ) = 1 | G ∩ [ N ] | (cid:88) n ∈ G ∩ [ N ] S ( n ) ≥| G ∩ [ M (cid:98) N/M (cid:99)|| G ∩ [ N ] | · | G ∩ [ M (cid:98) N/M (cid:99) ] | (cid:88) n ∈ G ∩ [ M (cid:98) N/M (cid:99) ] S ( n ) = | G ∩ [ M (cid:98) N/M (cid:99)|| G ∩ [ N ] | E n ∈ G [ M (cid:98) N/M (cid:99) ] S ( n ) . Moreover,(18) | G ∩ [ M (cid:98) N/M (cid:99)|| G ∩ [ N ] | ≥ | G ∩ [ M (cid:98) N/M (cid:99)| M (cid:98) N/M (cid:99) · M (cid:98) N/M (cid:99)
N > | G ∩ [ M (cid:98) N/M (cid:99)| M (cid:98) N/M (cid:99) · M ( N/M − N > δ for sufficiently large N . Combining (16), (17) and (18), for sufficiently large N , E n ∈ G ∩ [ N ] S ( n ) > δ M | F | This implies d G ( S ) > . This finishes our proof. (cid:3) Remark . Lemma 5.2 implies that a multiplicatively syndetic set in N has positive upperdensity. On the contrary, it is not true that an additively syndetic set has positive uppermultiplicative density. For example, the set N + 1 is additively syndetic but has zero multi-plicative density with respect to any multiplicative Følner sequence.It is worth noting that in (11), one can not replace lim sup with lim inf . In fact, we have: Proposition 5.4.
There exists a subsemigroup G of ( N , × ) such that lim inf N →∞ | G ∩ [ N ] || G ∩ [2 N ] | = 0 . Proof.
We construct G by putting primes into G in an inductive way as follows. Let ≤ N Theorem 5.5 ([14]) . Let G be a cancelative commutative semigroup and let T , . . . , T (cid:96) becommuting measure preserving actions of G on the probability space ( X, B , µ ) . Then for every A ∈ B with µ ( A ) > , there exist δ > and a syndetic set S ⊂ G such that for every g ∈ S , µ ( A ∩ T − ,g A ∩ · · · ∩ T − (cid:96),g A ) > δ. Now Theorem 1.3 follows from the next theorem. Theorem 5.6. Let ( X, B , µ, T ) be a measure preserving ( N , × ) -system and A ∈ B with µ ( A ) > . Let ( G, × ) be a subsemigroup of ( N , × ) . Then for every (cid:96) ∈ N , lim sup N →∞ E n ∈ G ∩ [ N ] µ ( A ∩ T − n A ∩ . . . ∩ T − n (cid:96) A ) > . If lim inf N →∞ | G ∩ [ N ] | N > (i.e. G has positive lower density), then we have the stronger bound lim inf N →∞ E n ∈ G ∩ [ N ] µ ( A ∩ T − n A ∩ . . . ∩ T − n (cid:96) A ) > . Remark . Theorem 5.6 is no longer true if we replace the Følner sequence ([ N ]) N ∈ N byarbitrary additive Følner sequence on N . For example, take ( X, B , µ, T ) to be the ( N , × ) -system where X = T , µ is the Lebesgue measure and T n x := x + log n , and let A be asmall interval on X . Then there exist arbitrary long intervals [ M, N ) of integers such that DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 25 A ∩ T − n A = ∅ for all n ∈ [ M, N ) . Therefore we can find a Følner sequence Φ consisting oflonger and longer such intervals with the property that lim N →∞ E n ∈ Φ N µ ( A ∩ T n A ) = 0 . Proof. In view of Theorem 5.5, there exists δ > such that the set S = { n ∈ G : µ ( A ∩ T − n A ∩ . . . ∩ T − n (cid:96) A ) > δ } is syndetic in ( G, × ) . We then have E n ∈ G ∩ [ N ] µ ( A ∩ T − n A ∩ . . . ∩ T − n (cid:96) A ) ≥ δ E n ∈ G ∩ [ N ] S . Theorem 5.6 now follows from Lemma 5.2. (cid:3) We conclude this section by remarking that one can not replace the lim inf and lim sup inTheorem 5.6 with limits. This is because the limits may not exist as the following propositionshows. Proposition 5.8. There exists a measure preserving system ( N , × ) -system ( X, B , µ, T ) anda set A ∈ B such that (20) lim N →∞ E n ∈ [ N ] µ ( A ∩ T − n A ) does not exist.Proof. Let χ : N → S be the completely multiplicative function χ ( n ) = n i = e i log n for n ∈ N .It is well known that(21) lim N →∞ E n ∈ [ N ] χ ( n ) does not exist. Consider the ( N , × ) -system ( X = S , µ, ( T n ) n ∈ N ) where µ is the normalizedLebesgue measure on X and T n ( x ) := χ ( n ) x for x ∈ X and n ∈ N . We show that this systemsatisfies our condition. By contradiction, assume for every A ⊂ X measurable, the limit in(20) exists.Let A be an arbitrary measurable subset of X . Let H be the smallest closed subspace of L ( X ) that contains T n A for all n ∈ N and constant functions. By our assumption, for every g ∈ H , lim N →∞ E n ∈ [ N ] (cid:90) X A ( T n x ) g ( x ) dµ ( x ) exists. Let F = (cid:40) f ∈ L ( X ) : lim N →∞ E n ∈ [ N ] (cid:90) X A ( T n x ) · f ( x ) dµ ( x ) exists (cid:41) . It follows that H ⊂ F . On the other hand, for all g ∈ H ⊥ , we have (cid:90) X A ( T n x ) · g ( x ) µ ( x ) = 0 . Hence lim N →∞ E n ∈ [ N ] (cid:90) X A ( T n x ) · g ( x ) dµ ( x ) exists and is equal to . In particular, H ⊥ ⊂ F . Therefore F = L ( X ) . In conclusion, forevery A ∈ B and every f ∈ L ( X ) , the limit lim N →∞ E n ∈ [ N ] (cid:90) X A ( T n x ) · f ( x ) µ ( x ) exists. By approximating f with simple functions, we get for all f ∈ L ( X ) , the limit lim N →∞ E n ∈ [ N ] (cid:90) X f ( T n x ) · f ( x ) µ ( x ) exists. However, this is not true as the following example shows. Let f : X → C be definedby f ( x ) = x for x ∈ S . Then f ( T n x ) = χ ( n ) x . We then have (cid:90) X f ( T n x ) f ( x ) dµ = χ ( n ) . Hence the limit(22) lim N →∞ E n ∈ [ N ] (cid:90) X f · T n f dµ does not exist. We reach a desired contradiction. (cid:3) Remark . One may ask whether the limit in (20) might exist after replacing Cesàro averageswith a different averaging scheme (for example logarithmic averages). However, for any givenaveraging scheme, there exists a complete multiplicative function χ for which the limit in (21)does not exist. Hence we can carry a similar construction as in proof of Proposition 5.8 toshow that there exists a system and set A for which the limit in (20) with respect to such anaveraging scheme does not exist.5.3. Subordinated semigroups. In this section, we introduce the notion of subordinatedsemigroups of ( N k , ∗ ) and some basic properties of these semigroups for later uses. Definition 5.10 (Subordinated semigroups) . We say that a semigroup ( N k , ∗ ) is subordinated to the Euclidean norm (cid:107) · (cid:107) in N k if there exists a constant C > such that(23) (cid:107) n ∗ m (cid:107) ≤ C (cid:107) n (cid:107)(cid:107) m (cid:107) for all n, m ∈ N k . DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 27 We remark that (23) is equivalent of saying that sup n,m ∈ N k (cid:107) n ∗ m (cid:107)(cid:107) n (cid:107)(cid:107) m (cid:107) is finite. Also note that,up to modifying the constant C , in the definition of subordinated semigroup we may changethe Euclidean norm (cid:107) · (cid:107) by any other equivalent norm.The following lemma follows immediately from the definition. Lemma 5.11. Let ( N k , ∗ ) be a semigroup subordinated to the Euclidean norm in N k . Thenfor every m ∈ N k , there exists K = K ( m ) ∈ N such that for all N ∈ N and n ∈ N k with (cid:107) n (cid:107) ≤ N , we have that (cid:107) m ∗ n (cid:107) ≤ KN . A class of subordinated semigroups of particular interest for us is given by semigroups ( N k , ∗ ) whose operation is induced by the usual matrix multiplication in a subsemigroup of M d × d ( Z ) for some d ∈ N . Here M d × d ( Z ) denotes the set of d × d -matrices with integercoefficients. Definition 5.12. Let d, k ∈ N . We say that a semigroup ( N k , ∗ ) is induced by M d × d ( Z ) (via ψ ) if there exists a linear injection ψ : N k → M d × d ( Z ) (meaning that ψ is a linearfunction of N k in every entry) such that ψ ( N k ) is a semigroup of M d × d ( Z ) not contained in { A ∈ M d × d ( Z ) : det( A ) = 0 } and such that n ∗ m = ψ − ( ψ ( n ) ψ ( m )) . Lemma 5.13. A semigroup ( N k , ∗ ) induced by M d × d ( Z ) is subbordinated to the Euclideannorm in N k .Proof. We regard M d × d ( Z ) as a subset of R d , and let N ∈ M k,d ( R ) and M ∈ M d ,k ( R ) be such that ψ ( x ) = N x and ψ − ( y ) = M y for x ∈ N k , y ∈ ψ ( N k ) . For any matrix A ∈ M d ,d ( R ) , denote (cid:107) A (cid:107) = sup x ∈ R d \{ } (cid:107) Ax (cid:107) / (cid:107) x (cid:107) . We have that (cid:107) n ∗ m (cid:107) = (cid:107) ψ − ( ψ ( n ) · ψ ( m )) (cid:107) ≤ (cid:107) M (cid:107)(cid:107) ψ ( n ) · ψ ( m ) (cid:107) ≤ (cid:107) M (cid:107)(cid:107) ψ ( n ) (cid:107)(cid:107) ψ ( m ) (cid:107) ≤ (cid:107) M (cid:107)(cid:107) N (cid:107) (cid:107) n (cid:107)(cid:107) m (cid:107) . (cid:3) The next lemma states that a syndetic subset in a semigroup subordinated to the Euclideannorm has positive lower density. Lemma 5.14. Let ( N k , ∗ ) be a semigroup subordinated to the Euclidean norm in N k and ( Z, ∗ ) be a subsemigroup of ( N k , ∗ ) with d + ( Z ) := lim sup N →∞ (cid:12)(cid:12) [ N ] k ∩ Z (cid:12)(cid:12) N k = 1 . Let S ⊆ Z be a syndetic subset (with respect to the ∗ operation). Then, lim inf N →∞ E n ∈ [ N ] k ∩ Z S ( n ) > . Proof. Since S is syndetic, there exists a finite set F ⊂ Z such that Z = F − ∗ S . Thenfor every x ∈ Z we have that ≤ (cid:80) n ∈ F S ( n ∗ x ) . By Lemma 5.11, there exist K =max { K ( n ) : n ∈ F } and N := max { N ( n ) : n ∈ F } such that for all N ≥ N , m ∗ n ∈ f ([ KN ] k ) for all n ∈ [ N ] k and m ∈ F . So for all N ≥ N , ≤ lim inf N →∞ E n ∈ [ N ] k ∩ Z (cid:88) m ∈ F S ( m ∗ n ) = lim inf N →∞ (cid:88) m ∈ F E n ∈ [ N ] k ∩ Z S ( m ∗ n ) ≤ lim inf N →∞ N k (cid:88) n ∈ [ KN ] k ∩ Z S ( n ) , where in the last inequality we used the fact that the map n → m ∗ n is injective from Z to Z for all m ∈ Z . Therefore,(24) lim inf N →∞ E n ∈ [ mN ] k ∩ Z S ( n ) > K − k > . Now for all N ≥ KN , let M = [ N/K ] , then (24) implies that lim inf N →∞ E n ∈ [ N ] k ∩ Z S ( n ) ≥ lim inf N →∞ (cid:16) KMN (cid:17) k E n ∈ [ KM ] k ∩ Z S ( n ) > lim inf N →∞ (cid:16) KMN (cid:17) k K − k > lim inf N →∞ (cid:16) K − N (cid:17) k . So lim inf N →∞ E n ∈ [ N ] k ∩ Z S ( n ) ≥ K − k and we are done. (cid:3) Parametrized multiplicative functions. In this section, we introduce a class of func-tions that parametrize multiplicative subsemigroups of Q > . Definition 5.15 (Parametrized multiplicative function) . For k ∈ N , we say that f : N k → Q ≥ is a parametrized multiplicative function if ¯ d + (cid:0) { n ∈ N k : f ( n ) = 0 } (cid:1) = 0 and there exists an operation ∗ : N k × N k → N k so that ( N k , ∗ ) is a semigroup subordinatedto the euclidean norm in N k and f ( n ∗ m ) = f ( n ) f ( m ) for all n, m ∈ N k . We say that f is a commutative parametrized multiplicative function if onecan choose ( N k , ∗ ) to be a cancelative commutative semigroup. Remark . We could, of course, disallow functions f : N k → Q ≥ that take the value ,but this would forbid some of the natural examples we exhibit below. On the other hand,we are interested in functions that take values in the multiplicative group ( Q > , × ) , so as acompromise we allow parametrized multiplicative functions to take the value , as long asonly on a set of zero additive density. DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 29 This definition leads to a situation where we may have a function f : N k → R which is onlydefined on a full density subset D of N k ; in such cases we denote by lim inf N →∞ E n ∈ [ N ] k f ( n ) := lim inf N →∞ E n ∈ [ N ] k D ( n ) f ( n ) . We sometimes write f : ( N k , ∗ ) → Q ≥ for a parametrized multiplicative function to stressthe semigroup operation for which f is a multiplicative function.A typical example of a parametrized multiplicative function arising from a semigroup ( N k , ∗ ) induced by M d × d ( Z ) is the one induced by the determinant function. That is f : ( N k , ∗ ) → Q ≥ , n (cid:55)→ | det( ψ ( n )) | , or more generally n (cid:55)→ ξ ( | det( ψ ( n )) | ) for any multiplicative function ξ : | det( ψ ( N k )) | → Q ≥ . Here we point out that since ψ ( N k ) is not contained in { A ∈M d × d ( Z ) : det( A ) = 0 } , the polynomial det( ψ ( n )) is non trivial and therefore ¯ d (cid:0) { n ∈ N k :det( ψ ( n )) = 0 } (cid:1) = 0 . (To see this, note that for ( n , . . . , n k − ) ∈ N k − there are at most d positive integers n k ∈ N such that det( ψ ( n , . . . , n k − , n k )) = 0 .) Example . In the following, we present some examples of parametrized multiplicativefunctions.(1) Consider the function f given by f = det ◦ ψ , where ψ : N k → M × ( Z ) , ( n , n ) (cid:55)→ (cid:32) n − n n n (cid:33) is a linear injection. By Lemma 5.13, the semigroup induced by M × ( Z ) via ψ is subbordinated to the Euclidean norm in N . Thus we have that f ( n , n ) = n + n is a parametrized multiplicative function.(2) If k = a + b for some a, b ∈ N , then f ( n , n ) = k ( n + n ) is a parametrizedmultiplicative function, since f = det ◦ ψ with ψ : N → M × ( Z ) being the linearinjection ( n , n ) (cid:55)→ (cid:32) a − bb a (cid:33) (cid:32) n − n n n (cid:33) .(3) Similar to the previous examples, the linear injection ψ : ( n , n ) (cid:55)→ (cid:32) n − Dn n n (cid:33) for D ∈ Z gives us the paramatrized multiplicative function f ( n , n ) = | det ◦ ψ ( n , n ) | = | n + Dn | .(4) Given two parametrized multiplicative functions, f : ( N d , ∗ ) → Q ≥ and f : ( N d , ∗ ) → Q ≥ , we may define for (cid:96) , (cid:96) ∈ Z the function f : ( N d × N d , ∗ × ∗ ) → Q , suchthat f ( n , n ) = f (cid:96) ( n ) f (cid:96) ( n ) , if f ( n ) f ( n ) (cid:54) = 0 and f ( n , n ) = 0 otherwise.Since ( N d i , ∗ i ) , i = 1 , is subordinated to the Euclidean norm in N d i , we get that ( N d × N d , ∗ × ∗ ) is subordinated to the Euclidean norm in N d × N d . It is nothard to check that the upper density of { ( n , n ) ∈ N d × N d : f ( n , n ) = 0 } equals0. This implies that f is a parametrized multiplicative function. This also shows that /f is a parametrized multiplicative function. In particular, for any p, q ∈ Z the function f : N → Q > , ( m, n ) (cid:55)→ m p n q is aparametrized multiplicative function.(5) Let K be a finite field extension of Q and O K be the ring of integers of K . Let d = [ K : Q ] and B = { b , . . . , b d } be an integral basis of O K , i.e., every x ∈ O K can bewritten as x = n b + · · · + n d b d for some n , . . . , n d ∈ Z in a unique way. Let N K ( x ) denote the norm of x in K . Since N K ( x ) N K ( y ) = N K ( xy ) , the function f : N d → Q > given by f ( n , . . . , n d ) = (cid:12)(cid:12) N K ( n b + · · · + n d b d ) (cid:12)(cid:12) is a parametrized multiplicativefunction. We remark that in this example, f can be written as f = det ◦ ψ for somelinear injection ψ : Z d → M d × d ( Q ) .As a special case, let ω = e πi/ and K = Q ( ω ) . Then { , ω, ω } is an integral basisof O K . For every x = a + a ω + a ω and y = b + b ω + b ω with a , a , a , b , b , b ∈ Q , note that xy = ( a b + a b + a b ) + ( a b + a b + a b ) ω + ( a b + a b + a b ) ω = (cid:16) b b b (cid:17) · a a a a a a a a a · ωω . Therefore the function f ( a , a , a ) := | N K ( a + a ω + a ω ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det a a a a a a a a a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | ( a + a + a )( a + a + a − a a − a a − a a ) | = | a + a + a − a a a | is a parametrized multiplicative function.(6) Let p : Z → Z be the minimal polynomial of A ∈ M n × n ( Z ) and d = deg( p ) . Let ψ : N d → M n × n ( Z ) be such that ψ ( n , . . . , n d − ) := n I n + n A + · · · + n d − A d − (which is a linear injection since d is the degree of the minimal polynomial of A ).Since p ( A ) = 0 , ψ ( N d ) is a semigroup and ψ − : ψ ( N d ) → N is a well defined map. Sowe may denote n ∗ m := ψ − ( ψ ( n ) ψ ( m )) . Also note that ψ ( N d ) does not consists of only non-invertible matrices. Then thegroup ( N d , ∗ ) is induced by M n × n ( Z ) (via the map ψ ), and so | det ◦ ψ | : N d → Z is aparametrized multiplicative function.To illustrate this class of examples, consider the Fibonacci matrix A = (cid:32) (cid:33) .Its minimal polynomial is of degree 2 and then we can consider ψ : N → M × ( Z ) , DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 31 given by ψ ( n, m ) = nI + mA = (cid:32) n + m mm n (cid:33) Hence | det ◦ ψ ( n, m ) | = | n − m + nm | is a parametrized multiplicative function.More generally, consider a matrix A = (cid:32) a bc d (cid:33) not being a multiple of the identity.Then its minimal polynomial is of degree 2 and therefore | det( nI + mA ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:32) n + am bmcm n + dm (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | n + ( a + d ) mn + ( ad − bc ) m | is a parametrized multiplicative function.(7) Consider the quaternion semigroup of matrices { M n ,n ,n ,n : n , n , n , n ∈ N } where M n ,n ,n ,n = n − n − n − n n n − n n n n n − n n − n n n . Via the linear injection ψ : N → M × ( Z ) with ( n , n , n , n ) (cid:55)→ M n ,n ,n ,n we ob-tain the parametrized multiplicative functions f ( n , n , n , n ) = det ◦ ψ ( n , n , n , n ) =( n + n + n + n ) and g ( n , n , n , n ) = (det) / ( ψ ( n , n , n , n )) = n + n + n + n .We remark that the functions in examples (1)–(6) are all commutative parametrized multi-plicative functions, however it is not clear whether the one in example (7) is, as the quaternionsemigroup is non-commutative.5.5. Averages along parametrized multiplicative functions. In this section we proveTheorem 1.4. In fact we will establish the following more general version which also deals withcommutative parametrized functions that take values in Q ≥ . Recall the convention adoptedin Remark 5.16. Theorem 5.18. Let ( X, B , µ, T ) be a measure preserving ( N , × ) -system and A ∈ B with µ ( A ) > . Let f : N k → Q ≥ be a commutative parametrized multiplicative function and (cid:96) ∈ N . Then lim inf N →∞ E n ∈ [ N ] k µ (cid:16) A ∩ T − f ( n ) A ∩ T − f ( n ) A ∩ · · · ∩ T − f ( n ) (cid:96) A (cid:17) > . Proof. The proof resembles that of Theorem 5.6. Let ( N k , ∗ ) be the semigroup that givesrise to the parametrized multiplicative function f . Let Z be the set of all n ∈ N k with f ( n ) > . Since f is a homomorphism, ( Z, ∗ ) is a subsemigroup of ( N k , ∗ ) . For each n ∈ Z , let R n := T f ( n ) . Denote by n ∗ k the iterate product n ∗ k = n ∗ n · · · ∗ n ( k -times). By Theorem 5.5,there exists δ > such that the set S = { n ∈ Z : µ ( A ∩ R − n A ∩ . . . ∩ R − n ∗ (cid:96) A ) > δ } is syndetic in ( Z, ∗ ) . Since d + ( Z ) = 1 by definition, we have lim inf N →∞ E n ∈ [ N ] k µ ( A ∩ T − f ( n ) A ∩ T − f ( n ) A ∩ · · · T − f ( n ) (cid:96) A )= lim inf N →∞ E n ∈ [ N ] k ∩ Z µ ( A ∩ T − f ( n ) A ∩ T − f ( n ) A ∩ · · · T − f ( n ) (cid:96) A ) ≥ δ lim inf N →∞ E n ∈ [ N ] k ∩ Z S ( n ) > , where the last inequality comes from Lemma 5.14. (cid:3) As an immediate corollary of Theorem 1.4, we conclude that for all multiplicative measurepreserving system ( X, B , µ, T ) and A ∈ B with µ ( A ) > , the following averages are positive:(1) lim inf N →∞ E m,n ∈ [ N ] µ ( T − m a A ∩ T − n b A ) for a, b ∈ N ; (2) lim inf N →∞ E n,m ∈ [ N ] µ ( A ∩ T − n + m A ) (which implies Proposition 1.5);(3) lim inf N →∞ E n,m ∈ [ N ] µ ( A ∩ T − n + Dm A ) for all D ∈ N ;(4) lim inf N →∞ E n,m ∈ [ N ] µ ( A ∩ T − k ( n + m ) A ∩ T ( k ( n + m )) A ) for all k of the form a + b for some a, b ∈ Z . Remark . We remark that item (4) above fails if k does not have the form a + b . Indeed,if this is the case, there exists a prime p ≡ such that p has odd exponent in primefactorization of k . For n ∈ N let f p ( n ) denote the exponent of p in the prime factorization of n . We then have f p ( k ( m + n )) is odd for all m, n ∈ N . For n ∈ N , color n with C ( n ) ≡ f p ( n )(mod 2) . It follows that { k ( m + n ) : m, n ∈ N } is not a set of topological multiplicativerecurrence.5.6. Single recurrence and non-commutative semigroups. In this section, we showthat for the special case (cid:96) = 1 of Theorem 1.4, one can drop the assumption that f arisesfrom a commutative semigroup. To this end, we need the following version of Khintchine’srecurrence theorem for general countable semigroups (not necessarily amenable) [2, Theorem5.1]). This result is probably well known; for completeness’ sake we provide a short proofwhich is almost verbatim taken from the proof of [2, Theorem 5.1]. Theorem 5.20 (Khintchine’s theorem) . Let G be a countable semigroup, let ( T g ) g ∈ G be anaction of G by measure preserving transformations on a probability space ( X, B , µ ) and let We do not know how to establish this without using Theorem 1.4 (i.e. using only Theorem 1.3). DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 33 A ∈ B with µ ( A ) > . Then for every (cid:15) > the set R := (cid:8) g ∈ G : µ ( A ∩ T − g A ) > µ ( A ) − (cid:15) (cid:9) is syndetic.Proof. Suppose, for the sake of a contradiction that R is not syndetic. Choose g ∈ G \ R arbitrary. Then, inductively, for each m ≥ let F m = (cid:8) g m g m − · · · g n : 1 ≤ n ≤ m (cid:9) andchoose g m +1 ∈ G \ RF − m . This is possible if R is not syndetic. From the construction wehave F m = g m F m − ∪ { g m } , and hence it follows that F m ∩ R ⊂ { g m } for every m ∈ N .Let N ∈ N be large, to be determined later depending only on µ ( A ) and (cid:15) and let f = N (cid:80) Nn =1 A ◦ T g n g n − ··· g . Clearly (cid:82) X f d µ = µ ( A ) . By the Cauchy-Schwarz inequality, µ ( A ) ≤ (cid:90) X f d µ = 1 N N (cid:88) n,m =1 µ (cid:0) T − g n g n − ··· g A ∩ T − g m g m − ··· g A (cid:1) = 1 N (cid:32) N (cid:88) n =1 µ ( A ) + 2 N (cid:88) m =2 m − (cid:88) n =1 µ (cid:0) T − g n g n − ··· g A ∩ T − g m g m − ··· g A (cid:1)(cid:33) = 1 N (cid:32) N µ ( A ) + 2 N (cid:88) m =2 m − (cid:88) n =1 µ (cid:0) A ∩ T − g m g m − ··· g n +1 g n A (cid:1)(cid:33) = 1 N µ ( A ) + 2 N N (cid:88) m =2 (cid:88) g ∈ F m µ (cid:0) A ∩ T − g A (cid:1) ≤ N µ ( A ) + 2 N (cid:18) N (cid:19)(cid:0) µ ( A ) − (cid:15) (cid:1) which is a contradiction if N is large enough. (cid:3) Theorem 5.21. Let ( X, B , µ, T ) be an ( N , × ) measure preserving system and A ∈ B with µ ( A ) > . Let f : ( N k , ∗ ) → Q ≥ be a parametrized multiplicative function. Then lim inf N →∞ E n ∈ [ N ] k µ ( A ∩ T − f ( n ) A ) > . Proof. Let ( N k , ∗ ) be the semigroup that gives the parametrized multiplicative function f .Let Z be the set of all n ∈ N k with f ( n ) > . Since f is a homomorphism, ( Z, ∗ ) is asubsemigroup of ( N k , ∗ ) . For n ∈ Z , denote R n := T f ( n ) . By Theorem 5.20, the set S := (cid:8) n ∈ Z : µ ( A ∩ R n A ) > µ ( A ) / (cid:9) is syndetic in ( Z, ∗ ) . Since µ ( A ∩ R n A ) ≥ µ ( A ) S ( n ) for all n ∈ Z and d + ( Z ) = 1 , itsuffices to show that lim inf N →∞ E n ∈ [ N ] k ∩ Z S ( n ) > . The last inequality follows from Lemma 5.14, so our proof finishes. (cid:3) Remark . As an immediate corollary of Theorem 5.21, we conclude that for all measurepreserving ( N , × ) -system ( X, B , µ, T ) and A ∈ B with µ ( A ) > , lim inf N →∞ E n ,n ,n ,n ∈ [ N ] µ ( A ∩ T − n + n + n + n A ) > . Density regularity of Pythagorean triples in finite fields Pythagorean pairs in finite fields. In what follows we explore the question of findingPythagorean pairs in sets of positive density in finite fields. We start with a result in numbertheory, which follows from [21, Theorem 5A] (see also the proof of Corollary 5B there). Theorem 6.1 ([21, Theorem 5A]) . Let k ∈ N , let F be a finite field, let Q = { x : x ∈ F } and let a , . . . , a k ∈ F be arbitrary and pairwise distinct. Then (cid:12)(cid:12)(cid:12) ( Q − a ) ∩ · · · ∩ ( Q − a k ) (cid:12)(cid:12)(cid:12) | F | = 12 k + o k, | F |→∞ (1) . In next proposition, we show that Conjecture 1.1 is true when N is replaced by a finite field.In fact, we prove a stronger result which involves “density regularity” rather than “partitionregularity”. Proposition 6.2. For every δ > there is M > such that whenever F is a finite field with | F | > M and A ⊂ F having | A | > δ | F | , there exist x, y ∈ A with x + y being a perfect square.Proof. Since | A | > δ | F | , the set B := { x : x ∈ A } has | B | > δ | F | .Our task is to show that there exist x, y ∈ B such that x + y is a perfect square when | F | issufficiently large. Denote by Q the set of perfect squares in F ; we will show that there exists x ∈ B such that ( Q − x ) ∩ B (cid:54) = ∅ . This will follow if we show that the set Q B := (cid:83) x ∈ B ( Q − x ) satisfies | Q B | > (1 − δ/ | F | .Let Q c := F \ Q and Q cB := F \ Q B . Then we have Q cB = (cid:84) x ∈ B ( Q c − x ) and we wantto show that | Q cB | < δ | F | . Let k ∈ N be such that / k < δ . In view of Theorem 6.1, | Q cB | = k + o k, | F |→∞ (1) , so if | F | is large enough we conclude that indeed | Q cB | < δ , asdesired. (cid:3) Remark . In contrast to Proposition 6.2, it is not true that every set of positive additivedensity of N contains x, y such that x + y is a perfect square. For example, consider the set N + 1 .6.2. Pythagorean triples in finite fields. It is also natural to ask whether an extensionof Proposition 6.2 stating that there exist x, y, z ∈ A with x + y = z still holds. We showthat this is false and characterize all equations of the form ax + by + cz = 0 for which theanalogous results are true. DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 35 Proposition 6.4. For a, b, c ∈ Z , the followings are equivalent:(i) For every δ > , there is M such that whenever F is a finite field with | F | > M and A ⊆ F with | A | > δ | F | , there exist x, y, z ∈ A such that ax + by + cz = 0 .(ii) a + b + c = 0 .Remark . We remark that related questions are also raised and addressed in [9, 19]. Proof. Suppose that a + b + c = 0 and F is a finite field. Let δ > and A ⊂ F with | A | > δ | F | .Set B := { x : x ∈ A } . Then | B | ≥ | A | / > δ/ | F | . It suffices to find x, t ∈ F such that x, x + ct, x − bt ∈ B , but this immediately follows from Szemerédi Theorem for finite groups([24, Theorem 10.5]).Now assume that a + b + c (cid:54) = 0 and let p > be a prime. Let m = | a + b + c | > and q bethe smallest prime such that q > max { m, } . For ≤ i ≤ m − and ≤ j ≤ q − , denote B i,j = (cid:110) x ∈ Z : iN/m ≤ x < ( i +1) p/m, x ≡ j (mod q ) and x is a perfect square (mod p ) (cid:111) and let A i,j = { x ∈ F : x ∈ B i,j } . Then (cid:80) m − i =0 (cid:80) p − j =0 | A i,j | = p .We say that ( i, j ) is good if mj (cid:54) = ip mod q . Let U be the set of ( i, j ) , ≤ i ≤ m − , ≤ j ≤ q − which is good, and V be the set of ( i, j ) which is not good. Using the estimation | A i,j | ≤ | B i,j | ≤ p/mq and the fact that | V | ≤ m , we have that (cid:88) ( i,j ) is good | A i,j | ≥ p − m − p/q. By the pigeonhole principle, if p is sufficiently large, then there exists a good pair ( i, j ) suchthat | A i,j | ≥ (1 − /q ) p/mq . It suffices to show that ax + by + cz (cid:54) = 0 for all x, y, z ∈ A i,j .Indeed, for any x, y, z ∈ B i,j , denote x (cid:48) = mx − ip , y (cid:48) = my − ip and z (cid:48) = mz − ip .Then ≤ x (cid:48) , y (cid:48) , z (cid:48) < p . If ax + by + cz = 0 mod p , then m ( ax + by + cz ) = 0 mod p .By the construction of x (cid:48) , y (cid:48) and z (cid:48) , we have that ax (cid:48) + by (cid:48) + cz (cid:48) = 0 mod p . Assume that ax + by + cz = kp and ax (cid:48) + by (cid:48) + cz (cid:48) = rp for some k, r ∈ Z , then ( mk − r ) p = ( a + b + c ) ip = mip . In other words, mk − r = mi . This implies that m | r . On the other hand, since − mp < ax (cid:48) + by (cid:48) + cz (cid:48) = rp < mp , we must have that r = 0 and ax (cid:48) + by (cid:48) + cz (cid:48) = 0 . Therefore, ax (cid:48) + by (cid:48) + cz (cid:48) ≡ ( mj − ip )( a + b + c ) = ± m ( mj − ip ) mod q. This is impossible since ( i, j ) is a good pair. This completes the proof. (cid:3) Remark . An example due to Frantzikinakis shows that there exists a subset of N of positivemultiplicative density that contains no triple x, y, z such that x + y = z . The example is asfollows: let ( T := R / Z , µ, T ) be the multiplicative measure preserving system where µ is theLebesgue measure and T n x = n x for all x ∈ T . Let x be a generic point for µ . Fix δ > small. Then the set E = { n ∈ N : n x mod 1 ∈ [1 / − δ, / δ ] } has multiplicative density δ > . For x, y, z ∈ E , ( x + y ) x mod 1 ∈ [0 , δ ] ∪ [1 − δ, and z x ∈ [1 / − δ, / δ ] .Hence ( x + y ) x (cid:54) = z x mod 1 . It follows that x + y (cid:54) = z .7. Open questions In this section, we collect some open questions regarding sets of multiplicative recurrencethat naturally arise from our study. Among other things, Theorem 1.6 shows that the set { ( n + 1) /n : n ∈ N } is a set of topological multiplicative recurrence. It is natural to askwhether it is a set of measurable recurrence. More ambitious one can ask: Question 7.1. Let ( X, B , µ, T ) be a multiplicative measure preserving system and A ∈ B with µ ( A ) > . Is it true that lim inf N →∞ E n ∈ [ N ] µ ( T − n A ∩ T − n +1 A ) > In [11, Problem 6], Frantzikinakis and Host ask whether the set { ( m + n ) /m : m, n ∈ N } is a set of topological multiplicative recurrence. In the spirit of this paper, we can ask theaverage version of this question, and some other related configurations: Question 7.2. Let ( X, B , µ, T ) be a multiplicative measure preserving system and A ∈ B with µ ( A ) > . Is it true that lim inf N →∞ E n,m ∈ [ N ] µ ( T − n + m A ∩ T − m A ) > , or lim inf N →∞ E n,m ∈ [ N ] µ ( T − n + n A ∩ T − m A ) > For a polynomial P ∈ Z [ x ] , Theorem 4.3 shows that { P ( n ) : n ∈ N } contains an infinitemultiplicative semigroup if and only if P ( x ) = ( ax + b ) d for some a, d ∈ N , b ∈ Z with a | b ( b − and d ∈ N . Here we can ask: Question 7.3. For which polynomial P ∈ Z [ x ] is the set { P ( n ) : n ∈ N } a set of topologicalmultiplicative recurrence?Even the answer for the following question is unknown: Question 7.4. Is the set { n + 1 : n ∈ N } a set of topological multiplicative recurrence?It is known that the set of shifted primes P − { p − p is a prime } and P + 1 = { p + 1 : p is a prime } are sets of additive recurrence. Hence it is of interest to ask: Question 7.5. Are the sets P − and P + 1 sets of multiplicative recurrence?We remark that neither of the sets P − and P + 1 contain a multiplicative semigroup,since for any a ∈ N we can factor a − a − a + 1) and a + 1 = ( a + 1)( a − a + 1) andhence P + 1 does not contain a perfect square and P − does not contain a perfect cube. DDITIVE AVERAGES OF MULTIPLICATIVE CORRELATION SEQUENCES AND APPLICATIONS 37 References [1] J. Auslander. Minimal flows and their extensions , volume 153 of North-Holland Mathematics Studies .North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. 12,13[2] V. Bergelson. Ergodic Ramsey theory–an update. In Ergodic theory of Z d actions , volume 228 of LondonMath. Soc. Lecture Note Ser. , pages 1–61. Cambridge Univ. Press, Cambridge, 1996. 32[3] V. Bergelson. The multifarious Poincaré recurrence theorem. In Descriptive set theory and dynamicalsystems , volume 277 of London Math. Soc. Lecture Note Ser. , pages 31–57. Cambridge Univ. Press,Cambridge, 2000. 9[4] V. Bergelson. Multiplicatively large sets and ergodic Ramsey theory. Israel J. Math. , 148:23–40, 2005. 6,7, 24[5] V. Bergelson. Combinatorial and diophantine applications of ergodic theory. In B. Hasselblatt and A. Ka-tok, editors, Handbook of Dynamical Systems , volume 1B, pages 745–841. Elsevier, 2006. 6[6] V. Bergelson. Questions on amenability. Enseign. Math. , 54(2):28–30, 2008. 9[7] V. Bergelson and R. McCutcheon. Recurrence for semigroup actions and a non-commutative Schur theo-rem. In Topological dynamics and applications (Minneapolis, MN, 1995) , volume 215 of Contemp. Math. ,pages 205–222. Amer. Math. Soc., Providence, RI, 1998. 6, 16[8] N. Bogoliouboff and N. Kryloff. La théorie générale de la mesure dans son application à l’étude dessystèmes dynamiques de la mécanique non linéaire. Ann. of Math. (2) , 38(1):65–113, 1937. 9[9] P. Csikvári, K. Gyarmati, and A. Sárközy. Density and Ramsey type results on algebraic equations withrestricted solution sets. Combinatorica , 32(4):425–449, 2012. 35[10] P. Erdős and R. L. Graham. Old and new problems and results in combinatorial number theory , volume 28.Université de Genève, L’Enseignement Mathématique, Geneva, 1980. 1[11] N. Frantzikinakis and B. Host. Higher order Fourier analysis of multiplicative functions and applications. J. Amer. Math. Soc. , 30(1):67–157, 2017. 1, 2, 20, 36[12] H. Furstenberg. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic pro-gressions. J. d’Analyse Math. , 31:204–256, 1977. 2, 3[13] H. Furstenberg. Recurrence in ergodic theory and combinatorial number theory . Princeton UniversityPress, Princeton, N.J., 1981. 6[14] H. Furstenberg and Y. Katznelson. An ergodic Szemerédi theorem for IP-systems and combinatorialtheory. J. d’Analyse Math. , 45:117–168, 1985. 24[15] M. Heule, O. Kullmann, and V. Marek. Solving and Verifying the Boolean Pythagorean Triples Problemvia Cube-and-Conquer , pages 228–245. Springer International Publishing, Cham, 2016. 1[16] T. Kamae and M. Mendès France. Van der Corput’s difference theorem. Israel J. Math. , 31(3-4):335–342,1978. 3, 16[17] O. Klurman and A. Mangerel. Rigidity theorems for multiplicative functions. Mathematische Annalen ,372:651–697, 2018. 4[18] I. Kříž. Large independent sets in shift-invariant graphs: solution of Bergelson’s problem. Graphs Combin. ,3(2):145–158, 1987. 9[19] S. Lindqvist. Partition regularity of generalised Fermat equations. Combinatorica , 38(6):1457–1483, 2018.35 [20] A. Sárközy. On difference sets of sequences of integers. I. Acta Math. Acad. Sci. Hungar. , 31(1–2):125–149,1978. 3[21] W. Schmidt. Equations over finite fields. An elementary approach . Lecture Notes in Mathematics, Vol.536. Springer-Verlag, Berlin-New York, 1976. 34[22] W. Sun. A structure theorem for multiplicative functions over the Gaussian integers and applications. J.Anal. Math. , 134(1):55–105, 2018. 1[23] W. Sun. Sarnak’s conjecture for nilsequences on arbitrary number fields and applications.http://arxiv.org/abs/1902.09712, 2019. 1[24] T. Tao and V. Vu. Additive Combinatorics . Cambridge University Press, 2006. 35 Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Univer-sidad de Chile & UMI-CNRS 2807, Beauchef 851, Santiago, Chile. Email address : [email protected] Department of Mathematics, Ohio State University, 231 W. 18th Ave., Columbus, OH 43210 Email address : [email protected] Mathematics Institute, University of Warwick, Coventry, UK Email address : [email protected] Department of Mathematics, Virginia Polytechnic Institute and State University, 225 StangerStreet, Blacksburg VA, 24061-1026, USA Email address ::