Quantitative twisted patterns in positive density subsets
QQUANTITATIVE TWISTED PATTERNS IN POSITIVE DENSITY SUBSETS
KAMIL BULINSKI AND ALEXANDER FISH
Abstract.
We make quantitative improvements to recently obtained results on the structure of the imageof a large difference set under certain quadratic forms and other homogeneuous polynomials. Previous proofsused deep results of Benoist-Quint on random walks in certain subgroups of SL r ( Z ) (the symmetry groups ofthese quadratic forms) that were not of a quantitative nature. Our new observation relies on noticing thatrather than studying random walks, one can obtain more quantitative results by considering polynomialorbits of these group actions that are not contained in cosets of submodules of Z r of small index. Our mainnew technical tool is a uniform Furstenberg-S´ark¨ozy theorem that holds for a large class of polynomials notnecessarily vanishing at zero, which may be of independent interest and is derived from a density incrementargument and Hua’s bound on polynomial exponential sums. Introduction
We begin by recalling the following result of Magyar on the abundance of distances in positive density subsetsof Z d , for d ≥ Theorem 1.1 (Magyar [9]) . For all (cid:15) > d ≥ k = k ( (cid:15), d ) > B ⊂ Z d has upper Banach density d ∗ ( B ) := lim L →∞ max x ∈ Z d | B ∩ ( x + [0 , L ) d ) | L d > (cid:15), then there exists a positive integer N = N ( B ) such that k Z >N ⊂ (cid:8) (cid:107) b − b (cid:107) | b , b ∈ B (cid:9) . It was realised in a series of works initiated by Bj¨orklund and the authors (see [2], [1]) that a similar resultholds if one replaces the Euclidean squared distance (cid:107) · (cid:107) with other quadratic forms or other generalfunctions. Before we state the general results, let us focus on some simple examples to demonstrate the typeof questions of interest. Theorem 1.2 ([2]) . Let F : Z → Z be the non-positive definite quadratic form F ( x, y, z ) = xy − z or F ( x, y, z ) = x + y − z . Then for all B ⊂ Z of positive upper Banach density there exists a positive integer k such that k Z ⊂ F ( B − B ) . Note however that unlike for the case of (cid:107) · (cid:107) in Magyar’s theorem, it was not established in these works that k depends only on the upper Banach density of B . One of the main purposes of this paper is to demonstratethat this integer k does indeed depend only on d ∗ ( B ) and not on B , and thus answering affirmatively aquestion posed in [5]. a r X i v : . [ m a t h . D S ] F e b KAMIL BULINSKI AND ALEXANDER FISH
Theorem 1.3.
Let F : Z → Z be the non-positive definite quadratic form F ( x, y, z ) = xy − z or F ( x, y, z ) = x + y − z . Then for all (cid:15) > k = k ( (cid:15) ) ∈ Z > such that for all B ⊂ Z with d ∗ ( B ) > (cid:15) we have that k Z ⊂ F ( B − B ) . Recall that a set A ⊂ Z r is called recurrent or a set of recurrence if for all measure preserving actions T : Z r (cid:121) ( X, µ ) and B ⊂ X with µ ( B ) > µ ( T a B ∩ B ) > a ∈ A. By Furstenberg’s correspondence principle [6], if A is recurrent then for all B ⊂ Z r with d ∗ ( B ) > B − B ) ∩ A (cid:54) = ∅ . For example, the Furstenberg-S´ark¨ozy theorem states the the set of squares is a setof recurrence. Thus it makes sense for us to make the following convenient definition. Definition 1.4.
We say that a function F : Z d → S has virtually recurrent level sets if it satisfies thefollowing condition: For all measure preserving actions T : Z d (cid:121) ( X, µ ) and B ⊂ X with µ ( B ) > k such that for all s ∈ F ( k Z d ) there exists v ∈ Z d with F ( v ) = s and µ ( B ∩ T v B ) > . We say that F : Z d → S has uniformly virtually recurrent level sets if for all (cid:15) > k = k ( (cid:15) ) > T : Z d (cid:121) ( X, µ ) and B ⊂ X with µ ( B ) > (cid:15) we have that for all s ∈ F ( k Z d ) there exists a v ∈ Z d such that F ( v ) = s and µ ( B ∩ T v B ) > . By Furstenberg’s correspondence principle we have the following combinatorial consequences of having vir-tually recurrent level sets.
Proposition 1.5.
A function F : Z d → S has virtually recurrent level sets if for all sets B ⊂ Z d with d ∗ ( B ) > k such that F ( k Z d ) ⊂ F ( B − B ). A function F : Z d → S hasuniformly virtually recurrent level sets if for all (cid:15) > k = k ( (cid:15) ) > F ( k Z d ) ⊂ F ( B − B ) . Thus Theorem 1.2 is a consequence of the statement that the maps F ( x, y, z ) = xy − z and F ( x, y, z ) = x + y − z have virtually recurrent level sets, which was shown in [2] and [1]. While our new resultTheorem 1.3 is a consequence of the statement that these maps have uniformly virtually recurrent level sets. Remark 1.6.
In order to be able to say that (cid:107) · (cid:107) has uniformly virtually recurrent level sets, one wouldneed to remove a finite set of exceptions, i.e., replace the condition F ( k Z d ) ⊂ F ( B − B ) with F ( k Z d ) ⊂ F ( B − B ) ∪ A B where A B is a finite set (depending on B ) as per Magyar’s theorem (see the Ergodicformulation given by the first author in [4]). However the examples that we study in this paper do notrequire this hence we avoid it.To motivate the use of this terminology note that if we could take k = 1 then this would mean that for each s in the range of F , the level set { v ∈ Z d | f ( v ) = s } is recurrent. On the other hand, if k = k ( B ) (cid:54) = 1 thenwe can only say that those level sets with values in the image of a finite index subgroup of Z d , namely k Z d ,are recurrent w.r.t to B (note that k could be different for each B so it is not quite correct to say that thoseparticular level sets are recurrent). But in geometric group theory the adverb virtually means up to a finiteindex subgroup . Remark 1.7.
Every finite index subgroup W ≤ Z d of index k = | Z d /W | contains k (cid:48) Z d for some k (cid:48) ≤ k d ,so in fact this further justifies our use of the word virtual (i.e., we did not need to restrict our attention tojust those subgroups of the form k Z d , but it is convenient to do so).To provide a non-example, consider the linear function F : Z → Z given by F ( x , x ) = x + x and considerthe Bohr set B = B ( θ, (cid:15) ) × B ( θ, (cid:15) ) ⊂ Z , where B ( θ, (cid:15) ) = { x ∈ Z | xθ ∈ ( − (cid:15), (cid:15) ) (mod 1) } , for some irrational θ ∈ T = R / Z and small enough (cid:15) >
0. Then d ∗ ( B ) = 4 (cid:15) > F ( B − B ) = B ( θ, (cid:15) ) − B ( θ, (cid:15) ) + B ( θ, (cid:15) ) − B ( θ, (cid:15) ) ⊂ B ( θ, (cid:15) )is contained in another Bohr set, which cannot contain a non-trivial subgroup of Z whenever θ is irrationaland (cid:15) < , as k Z θ is dense in T for all non-zero integers k . On the other hand, a theorem of Bogolyubov [3](see also [10]) states that for any B ⊂ Z of positive upper Banach density, the set B − B + B − B containsa Bohr set.1.1. Recurrent orbits.
The strategy in [2] for establishing that a function F has virtually recurrent levelsets involved studying the orbits of the linear automorphism group of F and taking advantage of the factthat - for the functions F of interest - this group has a rich enough algebraic structure (in particular, thataction on R d is irreducible), which enabled the use of deep results of Benoist-Quint. This motivates us tomake the following convenient definition. Definition 1.8.
A semigroup action Γ (cid:121) Z d is said to have virtually recurrent orbits if for all measurepreserving actions T : Z d (cid:121) ( X, µ ) and B ⊂ X with µ ( B ) > k such that forall v ∈ Z d there exists a γ ∈ Γ such that µ ( B ∩ T kγv B ) > . On the other hand, if A ⊂ Z d we say that a semigroup action Γ (cid:121) Z d has uniformly virtually recurrentorbits across A if for each (cid:15) > k = k ( (cid:15) ) such that for all measure preservingactions T : Z d (cid:121) ( X, µ ) and B ⊂ X with µ ( B ) > (cid:15) there exists 0 < k ≤ k ( (cid:15) ) such that for each v ∈ A thereexists γ ∈ Γ such that µ ( B ∩ T kγv B ) > . To explain the relationships between these concepts, suppose that Γ ≤ GL d ( Z ) is the group of linear auto-morphisms of F , then for each fixed v ∈ Z d the orbit k Γ v lies in the level set F − ( F ( kv )). Thus if Γ (cid:121) Z d has virtually recurrent orbits then F has virtually recurrent level sets. As for uniformly virtually recurrentorbits, there is the subtlety that k is only uniformly bounded but not necessarily the same for all B with µ ( B ) > (cid:15) , nonetheless we still recover a constant k in the definition of uniformly virtual recurrent level setsby considering k ! (as F ( k ! Z d ) ⊂ F ( k Z d ) for all k ≤ k ). Unfortunately, for the examples of interest wehave not been able to establish uniform virtual recurrence of all orbits, however we will show that we canfor at least one orbit in each level set, which implies virtually recurrent level sets (each level set may be adisjoint union of many different orbits that we cannot all control).Using this language, we can restate the main theorem in [2] as follows. KAMIL BULINSKI AND ALEXANDER FISH
Theorem 1.9 ([2]) . Let sl n ( Z ) ∼ = Z n − denote the additive group of n × n integer matrices of trace zero.Let Γ = SL n ( Z ) act on sl n ( Z ) by conjugation (the adjoint representation). Then this action has virtuallyrecurrent orbits. In particular as conjugation preserves determinants, the characteristic polynomial map sl n ( Z ) → Z [ t ] given by A (cid:55)→ det( tI − A ) has virtually recurrent level sets. Likewise, the map F : Z → Z given by F ( x, y, z ) = xy − z has virtually recurrent level sets (as stated in concrete terms in Theorem 1.2above), as can be seen by identifying ( x, y, z ) ∈ Z with (cid:34) z − yx − z (cid:35) ∈ sl ( Z )and noting that the determinant is F .We now provide a quantitative improvement of this result by demonstrating that this characteristic polyno-mial map is uniformly virtually recurrent, answering a question raised in [5]. To do this, we show that thisaction of SL n ( Z ) by conjugation has uniformly virtually recurrent orbits across the companion matrices ofcharacteristic polynomials. Definition 1.10.
Given a polynomial p ( t ) = a + a t + a t + · · · + a n − t n − + t n , we can define its companionmatrix by c p = . . . − a . . . − a ... . . . ... ... ...0 . . . − a n − . . . − a n − Theorem A.
Let Γ = SL n ( Z ) act on sl n ( Z ) by conjugation (the adjoint representation). Let A = { c p | p ( t ) ∈ Z [ t ] with p (cid:48) (0) = 0 } be the set of companion matrices of integer polynomials with zero linear term (so that the correspondingcompanion matrix has trace 0 and thus in sl n ( Z )). Then this conjugation action has uniformly recurrentorbits across A . Thus the characteristic polynomial map A (cid:55)→ det( tI − A ) has uniformly virtually recurrentlevel sets. Likewise, the quadratic form F ( x, y, z ) = xy − z has uniformly virtually recurrent level sets (as stated in simple terms in Theorem 1.3).We are able to recover this result by demonstrating that if the group is generated by unipotents and the orbitis not contained in any proper affine subspace (this holds for non-zero vectors and irreducible representationsin dimension greater than 1), then the orbit is virtually recurrent. Definition 1.11.
A set S ⊂ R r is said to be hyperplane-fleeing if for all proper affine subspaces H of R d (i.e., H = W + a for some proper vector subspace W ⊂ R r and a ∈ R r ) we have that S (cid:54)⊂ H . Theorem 1.12.
Suppose that Γ ≤ GL r ( Z ) is a subgroup generated by a finite set of unipotents such thatfor each non-zero v ∈ Z r the orbit Γ v is hyperplane-fleeing (this holds if r > (cid:121) R r is irreducible). Then this action has virtually recurrent orbits. Our next main result provides a quantitative improvement by demonstrating that if the orbit also flees cosetsof subgroups of large index sufficiently quickly, then we have uniform virtual recurrence.
Definition 1.13.
Let Λ be an abelian group. We say that S ⊂ Λ is Q -coset fleeing (in Λ ) if for all subgroups W ≤ Λ with index | Λ /W | > Q we have that S is not contained in a coset a + W of W , i.e., the image of S in the quotient Λ /W contains at least two elements. Theorem B.
Let (cid:15) > r, N, Q be positive integers. Then there exists a positive integer k = k ( r, N, Q, (cid:15) ) such that for all measure preserving systems T : Z r (cid:121) ( X, µ ) and B ⊂ X with µ ( B ) > (cid:15) there exists 0 < k ≤ k such that the following holds: Suppose that there exist N unipotent elements u , . . . , u N ∈ SL r ( Z ) and v ∈ Z r such that S = { u n · · · u n N N | n , . . . , n N ∈ Z } satisfies the property that S v is Q -coset fleeing in Z r and S v is also hyperplane-fleeing. Then µ ( B ∩ T kγv B ) > γ ∈ S . Thus, the results stated above on the uniformity for the adjoint representation will follow from establishinguniform bounds on N and Q for the orbits of the companion matrices.1.2. A uniform Furstenberg-S´ark¨ozy theorem.
Our main technical tool, which may be of independentinterest, is a quantitative uniform Furstenberg-S´ark¨ozy theorem that works for a large family of polynomialswhich do not necessarily have zero constant term.
Definition 1.14.
We say that a vector of integer coefficient polynomials P ( n ) = ( P ( n ) , . . . , P r ( n )) where P i ( n ) ∈ Z [ n ] has multiplicative complexity Q if for all (cid:126)a = ( a , . . . , a r ) ∈ Z r and q ∈ Z with gcd( a , . . . , a r , q ) =1 we have that the polynomial D (cid:88) j =1 b j n j = ( P ( n ) − P (0)) · (cid:126)a satisfies gcd( b , . . . , b D , q ) ≤ Q . Remark 1.15.
For the case r = 1, a polynomial P ( n ) = (cid:80) Dj =0 c j D j has multiplicative complexity gcd( c , . . . , c D ). Theorem C.
Let
D, r, Q be positive integers and (cid:15) >
0. There exists a positive integer k = k ( D, r, Q, (cid:15) )such that the following is true: Let T : Z r (cid:121) ( X, µ ) be an ergodic measure preserving system and supposethat B ⊂ X with µ ( B ) > (cid:15) . Then there exists a positive integer k = k ( B ) ≤ k such that whenever P : Z → Z r is a polynomial P ( n ) = ( P ( n ) , . . . , P r ( n )) with degree at most D (that is, P i ( n ) ∈ Z [ n ] with deg ( P i ) ≤ D ) such that P ( n ) has multiplicative complexity Q and P is hyperplane-fleeing, then there existsarbitrarily large n ∈ Z such that µ ( T kP ( n ) B ∩ B ) > . Note that if we were to add the condition P (0) = 0 then the result follows from the Furstenberg-S´ark¨ozytheorem with k = 1. Moreover, if we were allowed to change k (and even insist the bound k ≤ (cid:15) − + 1) fordifferent polynomials P then the result trivially follows from the Poincar´e Recurrence theorem. We cannotafford these relaxations in the proof of Theorem B from Theorem C given in Section 5. KAMIL BULINSKI AND ALEXANDER FISH
Remark 1.16.
We now demonstrate that we can not remove in Theorem C the assumption that P ( n ) hasmultiplicative complexity Q . To show this, suppose for contradiction that we could. Thus if we fix a positiveinteger m ≥ < (cid:15) < m then this would mean that there exists a positive integer k such thatfor every measure preserving system T : Z (cid:121) ( X, µ ) we would have that whenever B ⊂ X with µ ( B ) = m we would have a positive integer k = k ( B ) ≤ k such that for all integers a , a with a (cid:54) = 0 we would have µ ( B ∩ T k ( a n + a ) B ) > n ∈ Z . That is, we have chosen to focus on Z systems and polynomials of degree 1. Let X = Z /mk Z with uniformprobability measure (Haar measure) and let T x = x + 1. Now let B = { , . . . , k − } ⊂ X , which hasmeasure exactly m . Now choose a = k m and thus for all integers n and 0 < k ≤ k we have T k ( a n + a ) B = T ka B. But since | X | ≥ k , we can choose a suitable a such that T ka B is disjoint from B , which is a contradiction.1.3. A polynomial Bogolyubov Theorem.
To demonstrate another application of our techniques, wecan obtain quantitative polynomial extension of Bogolyubov’s result on the linear image of a difference setmentioned above.
Theorem 1.17 (Polynomial Bogolyubov’s theorem) . Let (cid:15) >
0, and assume that R ( n ) ∈ Z [ n ] is a polynomialsuch that deg ( R ) ≥ R (0) = 0. Let E ⊂ Z with d ∗ ( E ) > (cid:15) . Then there exists k ≤ k ( (cid:15), R ) such that k Z ⊂ { x + R ( y ) | ( x, y ) ∈ E − E } . Acknowledgement:
The authors were supported by the Australian Research Council grant DP210100162.2.
Polynomial exponential bounds
Throughout this paper, we let e ( t ) = exp(2 πit ). We begin with a classical bound of Hua. Theorem 2.1 ([8], see also [7]) . For (cid:15) > d there exists a constant C d,(cid:15) such that if f = a + a x + · · · + a d x d ∈ Z [ x ] is a polynomial and q is a positive integer such that gcd ( a , . . . , a d , q ) = 1then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q q (cid:88) n =1 e (cid:18) f ( n ) q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C d,(cid:15) q (cid:15) − d . We deduce a straightforward higher dimensional generalization that will be useful for us.We let S = { z ∈ C | | z | = 1 } be the multiplicative group of unit complex numbers. A character is a grouphomomorphism χ : Z r → S , i.e., χ ( x + y ) = χ ( x ) χ ( y ). Note that the image of χ has exactly q elements ifand only if it is of the form χ ( x , . . . , x r ) = e (cid:18) q ( a x + · · · + a r x r ) (cid:19) where gcd ( a , . . . a r , q ) = 1. Proposition 2.2.
For r, D, Q > (cid:15) r,D,Q : N → R with lim q →∞ (cid:15) r,D,Q ( q ) = 0such that the following is true: Suppose that P : Z → Z r is a polynomial function of degree at most D ,more precisely P ( n ) = ( P ( n ) , . . . , P r ( n )) with P i ( n ) ∈ Z [ n ] with deg( P i ) ≤ D . Suppose that P ( n ) has multiplicative complexity Q . Then for all positive integers q and characters χ : Z r → S with image ofcardinality at least q we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q q (cid:88) n =1 χ ( P ( n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) r,D,Q ( q ) . Proof.
Suppose that χ has cardinality exactly q . Hence χ ( x , . . . , x r ) = e (cid:18) q ( a x + · · · + a r x r ) (cid:19) where gcd ( a , . . . a r , q ) = 1. Let (cid:126)a = ( a , . . . , a r ) and write P ( n ) · (cid:126)a = P (0) · (cid:126)a + D (cid:48) (cid:88) j =1 b j n j where D (cid:48) ≤ D . Let q (cid:48) = gcd ( b , . . . , b D (cid:48) , q ) ≤ Q . Then by the Hua bound we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q q (cid:88) n =1 χ ( P ( n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q q (cid:88) n =1 e (cid:18) q P ( n ) · (cid:126)a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q q (cid:88) n =1 e q D (cid:48) (cid:88) j =1 b j n j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q q (cid:88) n =1 e q/q (cid:48) D (cid:48) (cid:88) j =1 b j q (cid:48) n j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C D (cid:48) ,(cid:15) ( q/q (cid:48) ) (cid:15) − D (cid:48) ≤ C D,(cid:15) ( q/Q ) (cid:15) − D . (cid:3)
3. ( q, δ ) -Equidistributed sets. In this section, we introduce the notion of a ( q, δ )-equidistributed subset of an ergodic system, which wasinitially developed by the first author in [4] in order to obtain a quantitative ergodic version of Magyar’stheorem. We include it again for the sake of completeness.For the remainder of this section, let F N = [1 , N ] d ∩ Z d . Definition 3.1.
Let T : Z d (cid:121) ( X, µ ) be an ergodic measure preserving action. Then we say that B ⊂ X is( q, δ ) -equidistributed if for almost all x ∈ X we havelim n →∞ | F n | |{ a ∈ F n | T qa x ∈ B }| ≤ (1 + δ ) µ ( B ) . Definition 3.2 (Conditional probability and ergodic components) . If (
X, µ ) is a probability space and C ⊂ X is measurable with µ ( C ) > µ ( ·| C ) given by µ ( B | C ) = µ ( B ∩ C ) µ ( C ) . We note that if C is invariant under some measure preserving action, then µ ( ·| C ) isalso preserved by this action. If T : Z d (cid:121) ( X, µ ) is ergodic and k is a positive integer, then the action T k : Z d (cid:121) ( X, µ ) may not be ergodic; but it is easy to see that there exists a T k -invariant subset C ⊂ X such that the action of T k on C is ergodic (more precisely, µ ( ·| C ) is T k -ergodic) and the translates of C disjointly cover X (there are at most k d distinct translates, hence µ ( C ) ≥ k − d ). Note that the translates of KAMIL BULINSKI AND ALEXANDER FISH C also satisfy these properties of C . We call such a measure µ ( ·| C ) a T k -ergodic component of µ . It followsthat µ is the average of its distinct T k -ergodic components.We may now introduce our measure increment technique, which will be used to reduce our recurrencetheorems to ones which assume sufficient equidistribution. Lemma 3.3 (Ergodic measure increment argument) . Let δ, (cid:15) >
0, let q be a positive integer and let T : Z d (cid:121) ( X, µ ) be ergodic. If B ⊂ X with µ ( B ) > (cid:15) then there exists a positive integer k ≤ q log( (cid:15) − ) / log(1+ δ ) and a T k -ergodic component, say ν , of µ such that ν ( B ) ≥ µ ( B ) and B is ( q, δ )-equidistributed with respectto T k : Z d (cid:121) ( X, ν ).To study the limits appearing in Definition 3.1 we make use of the well known Pointwise Ergodic Theorem.
Proposition 3.4 (Pointwise Ergodic Theorem) . Let T : Z d (cid:121) ( X, µ ) be a measure preserving action. Thenfor all f ∈ L ( X, µ ) there exists X f ⊂ X with µ ( X f ) = 1 such thatlim N →∞ | F N | (cid:88) a ∈ F N f ( T a x ) → P T f ( x )for all x ∈ X f . Proof of Lemma 3.3. If B is ( q, δ ) equidistributed, then we are done. Otherwise, it follows from the PointwiseErgodic Theorem (applied to the action T q and the indicator function of B ) that there exists a T q -ergodiccomponent of µ , say ν , such that ν ( B ) ≥ (1 + δ ) µ ( B ). Continuing in this fashion, we may produce amaximal sequence of ergodic components ν , ν . . . , ν J of T q , T q , . . . T q J , respectively, such that ν j +1 ( B ) ≥ (1 + δ ) ν j ( B ). Clearly we must have (cid:15) (1 + δ ) J ≤ k = q J . (cid:3) We now turn to demonstrating the key spectral properties of a ( q, δ )-equidistribution set.
Definition 3.5 (Eigenspaces) . If T : Z d (cid:121) ( X, µ ) is a measure preserving action and χ ∈ (cid:99) Z d is a characteron Z d , then we say that f ∈ L ( X, µ ) is a χ -eigenfunction if T a f = χ ( a ) f for all a ∈ Z d . We let Eig T ( χ ) denote the space of χ -eigenfunctions and for R ⊂ (cid:99) Z d we letEig T ( R ) = Span { f | f ∈ Eig( χ ) for some χ ∈ R } L ( X,µ ) . In particular, we will be intersted in the sets R q = { χ ∈ (cid:99) Z d | χ q = 1 } and R ∗ q = R q \ { } , where q ∈ Z .Note that the spaces Eig T ( χ ) are orthogonal to each other and hence Eig T ( R ) has an orthonormal basisconsiting of χ -eigenfunctions, for χ ∈ R . Note also that Ergodicity implies that each Eig T ( χ ) is at most onedimensional. Proposition 3.6.
Let T : Z d (cid:121) ( X, µ ) be an ergodic measure preserving action and suppose that B ⊂ X is ( q, δ )-equidistributed. Let h ∈ L ( X, µ ) be the orthogonal projection of B onto Eig T ( R ∗ q ). Then P T q B = µ ( B ) + h and (cid:107) h (cid:107) ≤ (cid:112) (2 δ + δ ) µ ( B ) . Proof.
Note that Eig T ( R q ) = L ( X, µ ) T q . This, together with the ergodicity of T , shows that h = P T q B − µ ( B ). Now the pointwise ergodic theorem, applied to the action T q , combined with the( q, δ )-equidistribution of B immediately gives that (cid:107) h (cid:107) = (cid:107) P T q B (cid:107) − (cid:107) µ ( B ) (cid:107) ≤ (1 + δ ) µ ( B ) − µ ( B ) = (2 δ + δ ) µ ( B ) . (cid:3) A quantitative polynomial mean ergodic theorem and proof of Theorem C
Theorem 4.1.
Let
D, r, Q be positive integers and (cid:15) >
0. There exists a positive integer q = q ( D, r, Q, (cid:15) )such that the following is true: Let P : Z → Z r be a polynomial P ( n ) = ( P ( n ) , . . . , P r ( n )) with degree atmost D (that is, P i ( n ) ∈ Z [ n ] with deg ( P i ) ≤ D ) such that P ( n ) has multiplicative complexity Q and P ishyperplane-fleeing. Let T : Z r (cid:121) ( X, µ ) be an ergodic measure preserving system and suppose that B ⊂ X is ( q, δ )-equidistributed. Then lim sup N →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) µ ( B ) − N N (cid:88) n =1 T P ( n ) B (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ √ δ + (cid:15). (1)To prove this theorem, we first show the following. Lemma 4.2.
Suppose that T : Z d (cid:121) ( X, µ ) is a measure preserving system. Let f ∈ L ( X, µ ) be orthogonalto the rational Kronecker factor of (
X, µ, T ). Then for all polynomials P ( n ) , . . . P d ( n ) ∈ Z [ n ] such that nonon-trivial R -linear combination of them is constant we have thatlim N →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N N (cid:88) n =1 T P ( n ) f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 0 , where P ( n ) = ( P ( n ) , . . . , P d ( n )) . Proof.
By the spectral theorem, there exists a positive Borel measure σ on T d such that (cid:104) T v f, f (cid:105) = (cid:90) T d e ( (cid:104) v, θ (cid:105) ) dσ ( θ ) for all v ∈ Z d . In particular, for each character χ , written as χ ( v ) = e ( (cid:104) v, θ (cid:105) ) for some θ ∈ T d , we have that (cid:104) P Eig T ( χ ) f, f (cid:105) = σ ( { θ } )where P Eig T ( χ ) f denotes the orthogonal projection to f onto the χ -eigenfunctions (this follows from applyingthe Mean Ergodic Theorem to the unitary action χ ( v ) − T v since the χ ( v ) − T v invariant functions areprecisely the χ -eigenfunctions). Hence since f ∈ L Rat ( X, µ, T ) ⊥ we have that σ ( Q d / Z d ) = 0 and hence (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N N (cid:88) n =1 T P ( n ) f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:90) Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N (cid:88) n =1 e ( (cid:104) P ( n ) , θ (cid:105) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dσ ( θ )where Ω = T d \ ( Q d / Z d ) . We now claim that if θ ∈ Ω then the polynomial (cid:104) P ( n ) , θ (cid:105) / ∈ R + Q [ n ]. To see this,suppose for contradiction that it is not, thus q ( n ) = (cid:88) θ i ( P i ( n ) − P i (0)) ∈ Q [ n ] This follows from the fact that all finite dimensional representations of a finite abelian group can be decomposed into onedimensional representations. and thus the polynomials q ( n ) , P ( n ) − P (0) , . . . , P d ( n ) − P d (0) are linearly dependent over the real numbersand hence over the rationals (as they are all rational polynomials) but P ( n ) − P (0) , . . . , P d ( n ) − P d (0) arelinearly independent by assumption, so we have a linear combination q ( n ) = (cid:88) i θ (cid:48) i ( P i ( n ) − P i (0))with θ (cid:48) i ∈ Q . But by linear independence of P ( n ) − P (0) , . . . , P d ( n ) − P d (0) we must have θ i = θ (cid:48) i ∈ Q .This means we can apply Weyl’s polynomial equidistribution theorem to get thatlim N →∞ N N (cid:88) n =1 e ( (cid:104) P ( n ) , θ (cid:105) ) = 0 for all θ ∈ Ω . The dominated convergence theorem now completes the proof. (cid:3)
Proof of Theorem 4.1.
Let q be such that (cid:15) r,D,Q ( q ) < (cid:15) for all q ≥ q for an (cid:15) r,D,Q as in Proposition 2.2and set q = lcm { , . . . , q } . By Lemma 4.2, the left hand side of (1) remains unchanged if we replace B with P Rat B . We can write P Rat B = µ ( B ) + (cid:88) χ ∈ R ∗ q c χ ρ χ + (cid:88) χ ∈ Rat \ R q c χ ρ χ where ρ χ is a χ -eigenfunction of norm 1 and c χ ∈ C . From Proposition 3.6 we get that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N N (cid:88) n =1 T n (cid:88) χ ∈ R ∗ q c χ ρ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) χ ∈ R ∗ q c χ ρ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (2 δ + δ ) µ ( B ) ≤ δ. (2)Now if χ ∈ Rat \ R q then the cardinality of the image of χ is q (cid:48) ≥ q and the map n (cid:55)→ χ ( P ( n )) is q (cid:48) periodic,hence by Proposition 2.2 we get thatlim N →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N (cid:88) n =1 χ ( P ( n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q (cid:48) q (cid:48) (cid:88) n =1 χ ( P ( n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15). This implies thatlim sup N →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N N (cid:88) n =1 T P ( n ) (cid:88) χ ∈ Rat \ R q c χ ρ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim sup N →∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:88) χ ∈ Rat \ R q (cid:32) N N (cid:88) n =1 χ ( P ( n )) (cid:33) c χ ρ χ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:15) (cid:88) χ ∈ Rat \ R q c χ ≤ (cid:15) µ ( B ) ≤ (cid:15) . Finally, combining this estimate with (2) and using the triangle inequality gives the desired estimate (1). (cid:3)
The Cauchy-Schwartz inequality and Theorem 4.1 immediately give the following.
Theorem 4.3.
Let
D, r, Q be positive integers and (cid:15) >
0. There exists a positive integer q = q ( D, r, Q, (cid:15) )such that the following is true: Let P : Z → Z r be a polynomial ( P ( n ) = ( P ( n ) , . . . , P r ( n )) with degree atmost D (that is, P i ( n ) ∈ Z [ n ] with deg ( P i ) ≤ D ) such that P ( n ) has multiplicative complexity Q and P ishyperplane-fleeing. Let T : Z r (cid:121) ( X, µ ) be an ergodic measure preserving system and suppose that B ⊂ X is ( q, δ )-equidistributed. Then there exist arbitrarily large n ∈ Z such that µ ( T P ( n ) B ∩ B ) > µ ( B ) − (cid:15) − √ δ. Proof of Theorem C.
Take q = q ( D, r, Q, (cid:15) ) as in Theorem 4.3. Now apply the measure increment argument(Lemma 3.3) to pass to a T k -ergodic component ν of µ such that B is ( q, δ )-equidistributed where δ = (cid:15) and k ≤ k ( q, (cid:15) ) ≤ q log( (cid:15) − ) / log(1+ δ ) . So we may apply Theorem 4.3 to the T k : Z r (cid:121) ( X, ν ) and get thedesired conclusion (note that ν ( B (cid:48) ) > µ ( B (cid:48) ) > (cid:3) Constructing polynomials from unipotent elements (proof of Theorem B)
We now show how to prove Theorem B on uniform recurrence of unipotent actions from Theorem C onuniform polynomial recurrence. This will amount to constructing appropriate polynomials from sufficientlynice sets of unipotent elements as given in the hypothesis of Theorem B.
Proposition 5.1.
Let N , r and Q be positive integers and let D = D ( N, r ) = ( r + 1) N +1 − r −
1. Suppose u , . . . , u N ∈ SL r ( Z ) are unipotent elements and v ∈ Z r is such that S = { u n · · · u n N N | n , . . . , n N ∈ Z } satisfies the properties that S v is hyperplane-fleeing and Q -coset fleeing in Z r . Then there exists a polynomial P : Z → Z r with P ( n ) = ( P ( n ) , . . . , P r ( n )) where P i ( n ) ∈ Z [ n ] such that P ( n ) has degree at most D ,multiplicative complexity Q and the image { P ( n ) | n ∈ Z } is contained in S v and is hyperplane fleeing. Proof. As u i is unipotent, we have that the entries of the matrix u n i i are polynomials of degree at most r in n i . Thus S ( n , . . . , n N ) = u n · · · u n N N v = ( S ( n , . . . n N ) , . . . , S r ( n , . . . , n N ))with S i ( n , . . . , n N ) ∈ Z [ n , . . . , n N ]. Suppose that (cid:126)a = ( a , . . . a r ) ∈ Z r and q ∈ Z are such that gcd ( a , . . . , a r , q ) = 1. Let b = S (0 , . . . ,
0) be the constant term of S . Now consider the polynomial F ( n , . . . , n N ) = ( S ( n , . . . , n N ) − b ) · (cid:126)a and note that it has 0 constant term. Let q (cid:48) be the gcd of q and all the coefficients of F ( n , . . . , n N ). We claimthat q (cid:48) ≤ Q . To see this, let U = Z /q Z and let θ : Z r → U be the map given by θ ( x , . . . , x r ) = (cid:80) ri =1 a i x i and note that it is surjective since gcd ( a , . . . , a r , q ) = 1. The image of θ ◦ S is contained in q (cid:48) U + θ ( b ), asit is a polynomial which each coefficient dividing q (cid:48) , hence S v is contained in the coset θ − ( q (cid:48) U ) + b , thusfrom the assumption that it is Q -coset fleeing we get that | Z r /θ − ( q (cid:48) U ) | ≤ Q . However as θ is surjective wehave that | Z r /θ − ( q (cid:48) U ) | = | U/q (cid:48) U | = q (cid:48) where the last equality holds since q (cid:48) | q . Thus q (cid:48) ≤ Q as required.Now observe that the substitutions n j (cid:55)→ n ( r +1) j induce a map Z [ n , . . . , n N ] → Z [ n ] that is injective onthe monomials appearing in S ( n , . . . , n N ). Hence F ( n r +1 , n ( r +1) , . . . , n ( r +1) N ) ∈ Z [ n ] is a polynomialin n which has the same set of coefficients as F ( n , . . . , n N ). Thus P ( n ) = S ( n r +1 , n ( r +1) , . . . , n ( r +1) N )has degree at most D = ( r + 1) N +1 − r − Q , as required. Finally, P ( n ) = ( P ( n ) , . . . , P r ( n )) is hyperplane fleeing as otherwise some linear combination of the P i ( n ) is aconstant function, hence the constant polynomial, hence some non-trivial linear combination of the S i ( n ) isconstant, contradicting the assumption that S v is hyperplane-fleeing. (cid:3) Applications to the adjoint representation and the proof of Theorem A
We now demonstrate how to deduce Theorem A from Theorem B by showing how the hypothesis of Theo-rem B is satisfied by the companion matrices in sl d ( Z ). This technique will be easily generalized to SO( F )for the quadratic form F ( x, y, z ) = x + y − z .Let Λ = sl d ( Z ) ∼ = Z r , where r = d −
1, be the additive group of d × d integer matrices with zero trace andwe let Γ = SL d ( Z ) act on Λ by conjugation.Note that Γ is generated by finitely many unipotents u , . . . , u (cid:96) (the elementary matrices) and hence if weset Γ N = { u k · · · u k N N | k i ∈ Z } where we use cyclic notation u n = u n (mod (cid:96) ) then Γ = (cid:83) N ≥ Γ N with Γ N ⊂ Γ N +1 . Note that the image of u i in SL r ( Z ) (i.e., the map v (cid:55)→ u i vu − i ) is also unipotent since this mapping SL d ( Z ) → SL r ( Z ) is a grouphomomorphism and a polynomial map.Given a polynomial p ( t ) = a + a t + · · · + a n − t n − + t n , we can define its companion matrix by c p = . . . − a . . . − a ... . . . ... ... ...0 . . . − a n − . . . − a n − The characteristic polynomial of c p is p ( t ). Assume now that a n − = 0. Now consider the elementary matrix γ = and notice that v = γ c p − c p = . . . − δ n, . . . − . . . . . . ... ... ...0 . . . (3)is a non-zero constant independent of p . Proposition 6.1.
There exist constants
Q, N < ∞ such that for all companion matrices c p ∈ sl d ( Z ) the setΓ N c p is Q -coset fleeing and hyperplane-fleeing. Note that γ c p here means conjugation not matrix multiplication, as that is the group action of interest. Note also that − δ n, is the Kronecker delta (so − n = 2 and 0 otherwise) Proof. As v (cid:54) = 0 and the action of SL d ( Z ) on the R -vector space sl d ( R ) is an irreducible representation,we have that the R -span of the orbit Γ v is the whole sl d ( R ). It follows that Z -span of Γ v is a finiteindex subgroup W of sl d ( Z ), say of index Q . In fact, there is a large enough n so that the Z -span of thepartial orbit Γ n v is W (as finitely generated abelian groups are Noetherian). Now take N ≥ n such thatΓ n γ ⊂ Γ N . We claim that these N and Q , clearly constructed independently of c p , satisfy the claim. Tosee this, suppose that W ≤ Λ = sl d ( Z ) is a subgroup such that Γ N c p ⊂ W + a for some a ∈ Λ. Then W ⊃ Γ N c p − Γ N c p ⊃ Γ n ( γ c p − c p ) = Γ n v and thus | Λ /W | ≤ Q as required. Likewise, since the R -span of Γ N v is the whole of sl d ( R ), the sameargument also gives that Γ N c p cannot be contained in any translate of a strict subspace, thus showing thatΓ N c p is also hyperplane-fleeing. (cid:3) Quadratic form x − y − z and the completion of the proof of Theorem 1.3 Let F ( x, y, z ) = x − y − z and observe that F ( x, y, z ) = det (cid:32) z − ( x + y ) x − y − z (cid:33) . Hence we may regard F as the determinant map on the abelian subgroupΛ = (cid:40)(cid:32) a a a a (cid:33) ∈ sl ( Z ) | a ≡ a mod 2 (cid:41) . Now notice that the conjugation action ofΓ = (cid:42)(cid:32) (cid:33) , (cid:32) (cid:33)(cid:43) , preserves this additive subgroup and acts irreducibly on sl ( R ) (see Appendix A). Moreover, if we let γ = (cid:32) −
20 1 (cid:33) , a t = (cid:32) t (cid:33) , v = (cid:32) − − (cid:33) then we notice that we have the following analogue of identity (3) γ a t − a t = v for all t ∈ Z . Thus, by Theorem B, the same argument as in Proposition 6.1 applies to show that the action of Γ on Λhas uniformly virtually recurrent orbits across the set { a t | t ∈ Z } and thus that F : Z → Z has uniformlyvirtually recurrent level sets, as claimed in Theorem 1.3.8. Quantitative polynomial Bogolyubov’s theorem
We now prove the Polynomial Bogolyubov Theorem (Theorem 1.17). By use of Furstenberg’s correspondenceprinciple [6] it is enough to show the following result.
Theorem 8.1.
Let (cid:15) >
0, and R ( n ) = r D n D + · · · + r n ∈ Z [ n ] be a polynomial satisfying R (0) = 0and D = deg R ≥
2. There exists a positive integer k ( (cid:15), r D ) such that for every ergodic Z measure-preserving system ( X, µ, T ) and any measurable set B ⊂ X satisfying µ ( B ) > (cid:15) there exists a positiveinteger k ≤ k ( (cid:15), r D ) such that for every m ∈ Z we can find ( x, y ) ∈ Z with x + R ( y ) = km and µ ( B ∩ T ( x,y ) B ) > . Proof.
Fix such a B ⊂ X with µ ( B ) > (cid:15) and let δ = (cid:15) . We observe that solutions to x + R ( y ) = kc contain the curve P k,c ( n ) = ( kc − R ( kn ) , kn ) . Note that each P k,c is hyperplane fleeing (as P k,c ( n ) · ( a , a ) is a non-constant polynomial for all ( a , a ) ∈ R \ { (0 , } as D ≥
2) and has multiplicative complexity r D k D +1 , hence k − P k,c ∈ Z [ n ] has multiplicativecomplexity r D k D . In particular, P ,c ( n ) has multiplicative complexity r D hence by Theorem 4.3 (appliedwith Q = r D and r = 2) there exists a positive integer q = q ( D, , r D , (cid:15) ) such that if it were the case that B is ( q , δ )- equidistributed then µ ( T P ,c ( n ) B ∩ B ) > µ ( B ) − (cid:15) > n ∈ Z . Hence the theorem is true with k = 1 in this case. So now assume that B is not ( q , δ )- equidistributed.Then by a measure increment argument as given in Lemma 3.3 there exists a T q ergodic component,say ν , of µ such that ν ( B ) ≥ (1 + δ ) µ ( B ). By Theorem 4.3 now applied to ν there exists an integer q = q ( D, , r D q D , (cid:15) ) such that if B is ( q , δ ) equdistributed with respect to ν then ν (( T q ) q − P q ,c ( n ) B ∩ B ) > ν ( B ) − (cid:15) > n ∈ Z . Note that since ν ( B (cid:48) ) > ⇒ µ ( B (cid:48) ) > k = q . Hence assumenow that B is not ( q , δ )-equdistributed, thus there exists a T q -ergodic component, say ν of ν such that ν ( B ) ≥ (1 + δ ) ν ( B ) ≥ (1 + δ ) µ ( B ). Note that ν is a T q q ergodic component of µ and so we mayrepeat the same argument as before with q q in place of q etc. This procedure must eventually stop asafter j steps the ergodic component will have ν j ( B ) ≥ (1 + δ ) j µ ( B ) ≥ (1 + δ ) j (cid:15) and so the number of stepsis bounded as a function of (cid:15) , as required. The final value of k is then a product of at most j ( (cid:15) ) integersdepending only on (cid:15) and r D , hence depends only on (cid:15) and r D as required. (cid:3) Appendix A. Some algebraic facts
Lemma A.1.
Let Γ ≤ G ≤ GL n ( R ) be groups such that G is the Zariski closure of Γ. Suppose that ρ : G → GL d ( R ) is an irreducible representation such that ρ is a polynomial map. Then the restriction ρ | Γ : Γ → GL d ( R ) is also irreducible. Proof.
Suppose on the contrary that the restriction is reducible. This means that there exists a proper linearsubspace W ≤ R d and w ∈ W such that ρ (Γ) w ⊂ W . Let π : R d → R d /W denote the quotient map. Then P : G → GL d ( R ) given by P ( g ) = π ( ρ ( g ) w ) is a polynomial in g which vanishes for all g ∈ Γ. Since G isthe Zariski closure of Γ, we get that P also vanishes on G and hence ρ ( G ) w ⊂ W , which contradicts theirreducibility of ρ . (cid:3) Lemma A.2.
Let a, b ∈ Z \ { } be non-negative integers. Then the subgroupΓ = (cid:42)(cid:32) a (cid:33) , (cid:32) b (cid:33)(cid:43) is Zariski dense in SL ( R ). Proof.
Let U ( t ) = (cid:32) t (cid:33) . We wish to show that the Zariski closure of Γ contains U ( t ) and its transpose, for all t ∈ R , as thesegenerate SL ( R ). Now suppose that P : SL ( R ) → R is a polynomial map which vanishes on all of Γ . Then,in particular, the polynomial R : R → R given by R ( x ) = P ( U ( x )) vanishes on the infinite set a Z , and so R ( x ) is the zero polynomial. Hence P vanishes on U ( t ), for all t ∈ R . This shows that U ( t ) is in the Zariskiclosure, and a similar argument applies to its transpose. (cid:3) Example A.3.
The adjoint representation Ad : SL d ( R ) → GL( sl d ( R )) is a polynomial map. It is anirreducible representation and hence the above may be applied to verify the claims in Theorem 1.9. References [1] Bj¨orklund, M.; Bulinski, K.
Twisted patterns in large subsets of Z N . Comment. Math. Helv. 92 (2017), no. 3, 621–640.[2] Bj¨orklund, M.; Fish, A. Characteristic polynomial patterns in difference sets of matrices.
Bull. Lond. Math. Soc. 48 (2016),no. 2, 300–308.[3] Bogolio`uboff, N.
Sur quelques propri´et´es arithm´etiques des presque-p´eriodes.
Ann. Chaire Phys. Math. Kiev, 4:185–205,1939[4] Bulinski, K.
Spherical recurrence and locally isometric embeddings of trees into positive density subsets of Z d . Math. Proc.Cambridge Philos. Soc. 165 (2018), no. 2, 267–278.[5] Fish, A. On product of difference sets for sets of positive density.
Proc. Amer. Math. Soc. 146 (2018), no. 8, 3449–3453[6] Furstenberg, H.
Ergodic behavior of diagonal measures and a theorem of Szemer´edi on arithmetic progressions.
J. AnalyseMath. 31 (1977), 204–256.[7] Hua, L. K.
Additive theory of prime numbers.
Translations of Mathematical Monographs, Vol. 13 American MathematicalSociety, Providence, R.I. 1965 xiii+190 pp.[8] Hua, L. K.
On an exponential sum.
J. Chinese Math. Soc. 2 (1940), 301–312.[9] Magyar, ´A.
On distance sets of large sets of integer points.
Israel J. Math. 164 (2008), 251–263.[10] Ruzsa, I. Z.
Sumsets and structure.
In Combinatorial number theory and additive group theory, Adv. Courses Math. CRMBarcelona, pages 87–210. Birkh¨auser Verlag, Basel, 2009.
School of Mathematics and Statistics, University of Sydney, Australia
Email address : [email protected] School of Mathematics and Statistics, University of Sydney, Australia
Email address ::