Divisibility problems for function fields
aa r X i v : . [ m a t h . N T ] O c t DIVISIBILITY PROBLEMS FOR FUNCTION FIELDS
STEPHAN BAIER, ARPIT BANSAL, RAJNEESH KUMAR SINGH
Abstract.
We investigate three combinatorial problems considered by Erd¨os,Rivat, Sark¨ozy and Sch¨on regarding divisibility properties of sum sets and setsof shifted products of integers in the context of function fields. Our results inthis function field setting are better than those previously obtained for subsetsof the integers. These improvements depend on a version of the large sieve forsparse sets of moduli developed recently by the first and third-named authors.
Contents
1. Introduction and main results 12. Notations 33. Large sieve inequalities for function fields 53.1. Large sieve with additive characters 53.2. Large sieve with multiplicative characters 53.3. Arithmetic form of the large sieve 54. Proof of Theorem 1.4 95. Proof of Theorem 1.5 106. Proof of Theorem 1.6 11References 121.
Introduction and main results
In this paper, we investigate three combinatorial problems considered by Erd¨os,Rivat, Sark¨ozy and Sch¨on regarding divisibility properties of sum sets and setsof shifted products of integers in the context of function fields. The results weobtain in this function field setting are comparably better than those obtained inthe case of integers. We first state the best known results on these problems inthe integer setting.
Theorem 1.1 (Theorem in [1]) . We call a subset S of the positive integers a P -set if no element of S divides the sum of two ( not necessarily distinct ) largerelements of S . Let S be a P -set of pairwise coprime positive integers and denote Mathematics Subject Classification. by A S ( N ) the number of elements of S not exceeding N . If ε > , then thereexist infinitely many positive integers N such that A S ( N ) ≤ (3 + ε ) N / (log N ) − . Theorem 1.2 (Theorem 3 in [7]) . If N ∈ N is sufficiently large, A ⊆ { , ..., N } and a + a ′ is square-free for all a, a ′ ∈ A , then ♯ ( A ) ≤ N / log N. Theorem 1.3 (Theorem 4.2. in [10]) . If N ∈ N is sufficiently large, A , B ⊆{ , ..., N } and ab + 1 is square-free for all a ∈ A and b ∈ B , then min { ♯ ( A ) , ♯ ( B ) } ≤ N / log N. For the history of these problems, see the papers [1], [7] and [10]. It is likelythat the exponents in the estimates in the above theorems are not optimal. Wenote that the exponent 2 / /
2, however. In fact, Erd¨os and Sark¨ozy [7] gave a simple exampleof a P -set S satisfying A S ( N ) ≫ N / (log N ) − as N → ∞ , namely the set of squares of primes p ≡ P -set S satisfying A S ( N ) ≫ N / (log N ) − / (log log N ) − (log log log N ) − as N → ∞ . To date, this is the best known lower bound. It would be interestingto produce such lower bounds in the function field setting as well, but we willnot consider this problem here.The proofs of Theorems 1.1, 1.2 and 1.3 crucially depend on the large sieve.In all three proofs, the sets of relevant moduli in the large sieve are sparse, but itturns out that the results developed by L. Zhao and the first-named author of thispaper on the large sieve with sparse sets of moduli (see [2], [4] and [12]) are notsufficient to obtain any improvement over what can be established using the largesieve with full sets of moduli. However, the situation is different in the functionfield setting. Recently, the first and third-named authors of the present paperestablished an essentially best possible large sieve inequality with sparse setsof moduli for function fields (see [3]) which allows to obtain comparably betterresults for the analogues of the combinatorial problems above in the functionfield setting. This is the object of this paper. We prove the following resultscorresponding to Theorems 1.1, 1.2 and 1.3, where in the theorems below, F q denotes a finite field with q elements. Theorem 1.4.
We call a subset S of F q [ t ] a P -set if it consists of monic poly-nomials and no element of S divides the sum of two ( not necessarily distinct ) elements of S of larger degree. Let S be a P -set of pairwise coprime polynomials IVISIBILITY PROBLEMS FOR FUNCTION FIELDS 3 in F q [ t ] and denote by A S ( N ) the number of elements of S of degree not exceeding N . If ε > , then there exist infinitely many positive integers N such that A S ( N ) ≤ q N (1 / Φ+ ε ) , where Φ is the golden ratio, given by Φ := √ . We remark that the statement of Theorem 1.4 is empty if q = 2 n for some n ∈ N since in this case, 2 f = 0 for any polynomial f ∈ F q [ t ]. Theorem 1.5. If ε > , N ∈ N is sufficiently large, F is a set of monic polyno-mials in F q [ t ] of degree not exceeding N and f + f ′ is square-free for all f, f ′ ∈ F ,then ♯ ( F ) ≤ q N (2 / ε ) . Theorem 1.6. If ε > , N ∈ N is sufficiently large, F , G are sets of monicpolynomials in F q [ t ] of degree not exceeding N and f g + 1 is square-free for all f ∈ F and g ∈ G , then min { ♯ ( F ) , ♯ ( G ) } ≤ q N (2 / ε ) . In the last two theorems, “square-free” has the obvious meaning : A monicpolynomial in F q [ t ] is square-free if each factor appears precisely once in its uniquefactorization into irreducible monic polynomials. This is equivalent to saying thatthe polynomial has no multiple roots in the algebraic closure F q [ t ] since F q is aperfect field.To compare the above Theorems 1.4, 1.5 and 1.6 on sets of polynomials withTheorems 1.1, 1.2 and 1.3 on sets of integers, we note that the term q N in theestimates in Theorems 1.4, 1.5 and 1.6 takes the rule of N in the estimates inTheorems 1.1, 1.2 and 1.3. Thus, the exponent 2 / / Φ = 0 . ... in Theorem 1.4, and the exponent 3 / / Acknowledgement:
We would like to thank Igor Shparlinski for making usaware of Theorems 1.2 and 1.3 and suggesting to consider them in the functionfield setting. 2.
Notations
We begin by recalling some standard notations and facts about function fields.Let F q be a fixed finite field with q elements of characteristic p and letTr : F q → F p STEPHAN BAIER, ARPIT BANSAL, RAJNEESH KUMAR SINGH be the trace map. Let F q ( t ) ∞ be the completion of F q ( t ) at ∞ (i.e. F q ((1 /t ))).The absolute value | · | ∞ of F q ( t ) ∞ is defined by (cid:12)(cid:12)(cid:12)(cid:12) n X i = −∞ a i t i (cid:12)(cid:12)(cid:12)(cid:12) ∞ = q n , if 0 = a n ∈ F q . In particular, if f ∈ F q [ t ], then | f | ∞ = q deg f . The non-trivial additive character E : F q → C ∗ is defined by E ( x ) = exp (cid:26) πip Tr( x ) (cid:27) , and the map e : F q ( t ) ∞ → C ∗ is defined by e (cid:18) n X i = −∞ a i t i (cid:19) = E ( a − ) . This map e is a non-trivial additive character of F q ( t ) ∞ . Furthermore, the addi-tive characters modulo f ∈ F q [ t ] are precisely of the form ψ : F q [ t ] −→ C ∗ , ψ ( r ) = e (cid:18) r · gf (cid:19) , where g runs over a system of coset representatives of ( f ) in F q [ t ] (see [8]).In analogy to Dirichlet characters, we define multiplicative characters modulo f ∈ F q [ t ] to be multiplicative maps χ : F q [ t ] −→ C satisfying χ ( r ) = 0 if and only if ( r, f ) = 1and χ ( r ) = χ ( r ) if r ≡ r mod f. A multiplicative character modulo f ∈ F q [ t ] is referred to as primitive if it is notinduced by a multiplicative character modulo a polynomial g ∈ F q [ t ] of smallerdegree, i.e. if there does not exist a polynomial g ∈ F q [ t ] of smaller degree and amultiplicative character ˜ χ modulo g such that χ ( r ) = ˜ χ ( r ) whenever ( r, f ) = 1 . The principal character modulo f ∈ F q [ t ] is defined as χ ( r ) := ( r, f ) = 1 , r, f ) = 1 . IVISIBILITY PROBLEMS FOR FUNCTION FIELDS 5 Large sieve inequalities for function fields
Large sieve with additive characters.
Below we recall the large sieveinequality for dimension 1 from the recent paper [3] by the first and third-namedauthors. This will serve as the key tool in our present paper.
Theorem 3.1.
Let Q and N be positive integers, and S be a set of non-zeromonic polynomials in F q [ t ] of degree not exceeding Q . Then X f ∈ S X r mod f, ( r,f )=1 (cid:12)(cid:12)(cid:12)(cid:12) X g ∈ F q [ t ]deg g ≤ N a g e (cid:16) g · rf (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:0) q N +1 + ( ♯S ) · q Q − (cid:1) · X g ∈ F q [ t ]deg g ≤ N | a g | , where a g are arbitrary complex numbers. Large sieve with multiplicative characters.
Using Gauss sums, onecan turn the large sieve for additive characters into the following large sievefor multiplicative characters. The proof is completely analogue to the proof ofTheorem 7 in [5], which states the large sieve for multiplicative characters in theclassical case of integers.
Theorem 3.2.
Under the conditions of Theorem , we have X f ∈ S q deg f φ ( f ) X χ mod f,χ primitive (cid:12)(cid:12)(cid:12)(cid:12) X g ∈ F q [ t ]deg g ≤ N a g χ ( g ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:0) q N +1 + ( ♯S ) · q Q − (cid:1) · X g ∈ F q [ t ]deg g ≤ N | a g | , where a g are arbitrary complex numbers and f is the Euler function for F q [ t ] ,defined as φ ( f ) := ♯ { r mod f : ( r, f ) = 1 } . Proof. (cid:3)
Arithmetic form of the large sieve.
Similarly as in the case of integers,the large sieve with additive characters for function fields can be turned intoan upper bound sieve. The following result corresponds to Montgomery’s [9]arithmetic form of the large sieve in the classical setting of integers.
Theorem 3.3.
Let
Q < N be positive integers, assume that N ⊆{ f ∈ F q [ t ] : f monic , Q < deg f ≤ N } , P ⊆{ P ∈ F q [ t ] : P monic , deg P ≤ Q } such that P , P ∈ P = ⇒ ( P , P ) = 1 or P = P , Ω P := w ( P ) [ i =1 R i ( P ) if P ∈ P , where R i ( P ) ( i = 1 , ..., w ( P ))are distinct residue classes modulo P, STEPHAN BAIER, ARPIT BANSAL, RAJNEESH KUMAR SINGH and set N ∗ := { f ∈ N : f / ∈ Ω P for all P ∈ P } . Further, set w ( P ) = 0 if P P . Define the set K to be the union of { } and theset of all products of distinct elements of P , i.e. K : = { } ∪ { P · · · P n : n ∈ N , P , ..., P n are distinct elements of P } and A K ( Q ) = ♯ { k ∈ K : deg k ≤ Q } . Then N ∗ ≤ (cid:0) q N +1 + A K ( Q ) · Q q Q − (cid:1) (cid:18) X R ∈ F q [ t ] monicdeg R ≤ Q g ( R ) (cid:19) − , (1) where the function g : F q [ t ] → R is defined by g ( R ) := Y P | R w ( P ) q deg P − w ( P ) if R ∈ K and g ( R ) = 0 otherwise. To prove Theorem 3.3, we first introduce the following additional notations. If α ∈ F q ( t ) ∞ , we set S ( α ) := X g ∈ F q [ t ] a g e ( gα ) , where we put a g := 0 if deg g > N . We further set w ′ ( P ) := { h mod P : a g = 0 for all g ≡ h mod P } if P ∈ P . We define the function g ′ : F q [ t ] → R by g ′ ( R ) : = Y P | R w ′ ( P ) q deg P − w ( P ) (2)if R ∈ K and g ′ ( R ) = 0 otherwise. We first establish the following. Lemma 3.4.
In the above notations, we have g ′ ( R ) (cid:12)(cid:12)(cid:12)(cid:12) X g ∈ F q [ t ] a g (cid:12)(cid:12)(cid:12)(cid:12) ≤ X r mod RP ∤ r if P ∈ P and P | R (cid:12)(cid:12)(cid:12) S (cid:16) rR (cid:17)(cid:12)(cid:12)(cid:12) (3) for every R ∈ F q [ t ] .Proof. The said inequality is trivial if R
6∈ K . Therefore, we assume R ∈ K throughout the following. We note that X g ∈ F q [ t ] a g = S (0) , IVISIBILITY PROBLEMS FOR FUNCTION FIELDS 7 and therefore the claimed inequality (3) is equivalent to g ′ ( R ) · | S (0) | ≤ X r mod RP ∤ r if P ∈ P and P | R (cid:12)(cid:12)(cid:12) S (cid:16) rR (cid:17)(cid:12)(cid:12)(cid:12) . (4)Let β ∈ F q ( t ) ∞ be arbitrary. In the following, let us replace a g by a g e ( gβ ) in theoriginal definition of S . Then g ′ ( R ) doesn’t change and hence (4) gives us g ′ ( R ) · | S ( β ) | ≤ X r mod RP ∤ r if P ∈ P and P | R (cid:12)(cid:12)(cid:12) S (cid:16) rR + β (cid:17)(cid:12)(cid:12)(cid:12) (5)for all β ∈ F q ( t ) ∞ . Now suppose that (4) holds for R and R ′ with ( R, R ′ ) = 1.Then inequalities (4) and (5) above imply that the inequality (4) also holds for RR ′ in place of R because X r mod RR ′ P ∤ r if P ∈ P and P | RR ′ (cid:12)(cid:12)(cid:12) S (cid:16) rRR ′ (cid:17)(cid:12)(cid:12)(cid:12) = X a mod RP ∤ a if P ∈ P and P | R X a ′ mod R ′ P ∤ a ′ if P ∈ P and P | R ′ (cid:12)(cid:12)(cid:12)(cid:12) S (cid:16) aR + a ′ R ′ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≥ X a mod RP ∤ a if P ∈ P and P | R g ′ ( R ) · (cid:12)(cid:12)(cid:12) S (cid:16) aR (cid:17)(cid:12)(cid:12)(cid:12) ≥ g ′ ( R ) g ′ ( R ′ ) · | S (0) | = g ′ ( RR ′ ) · | S (0) | . By these considerations it suffices to prove (4) for for R = P ∈ P . To this end,we look at Z ( P, h ) := X g ∈ F q [ t ] g ≡ h mod P a g and Z := X g ∈ F q [ t ] a g = S (0) . (6)Opening up the square and using orthogonality relations for additive characters,it is easy to calculate that X r mod P (cid:12)(cid:12)(cid:12) S (cid:16) rP (cid:17)(cid:12)(cid:12)(cid:12) = q deg P X h mod P | Z ( P, h ) | . Subtracting | S (0) | = | Z | , we get X r mod PP ∤ r (cid:12)(cid:12)(cid:12) S (cid:16) rP (cid:17)(cid:12)(cid:12)(cid:12) = q deg P X h mod P | Z ( P, h ) | − | Z | . (7) STEPHAN BAIER, ARPIT BANSAL, RAJNEESH KUMAR SINGH
Finally, we use the equation Z = X h mod P Z ( P, h )and the Cauchy-Schwarz inequality to deduce that | Z | ≤ ( q deg P − w ′ ( P )) X h mod P | Z ( P, h ) | . Plugging this into (7) and using the definition of Z in (6) gives us the desiredinequality g ′ ( P ) (cid:12)(cid:12)(cid:12) X g ∈ F q [ t ] a g (cid:12)(cid:12)(cid:12) ≤ X r mod PP ∤ R (cid:12)(cid:12)(cid:12) S (cid:16) rP (cid:17)(cid:12)(cid:12)(cid:12) . (8) (cid:3) Now we are ready for the proof of Theorem 3.3.
Proof.
Summing (3) over R ∈ T with T := { k ∈ K : deg k ≤ Q } and noting that g ′ ( R ) = 0 if deg R ≤ Q and R
6∈ T , we obtain (cid:18) X R ∈ F q [ t ] monicdeg R ≤ Q g ′ ( R ) (cid:19) · (cid:12)(cid:12)(cid:12)(cid:12) X g ∈ F q [ t ] a g (cid:12)(cid:12)(cid:12)(cid:12) ≤ X R ∈T X r mod RP ∤ r if P ∈ P and P | R (cid:12)(cid:12)(cid:12) S (cid:16) rR (cid:17)(cid:12)(cid:12)(cid:12) = X D =1 monic D | R for some R ∈T X s mod D ( s,D )=1 (cid:12)(cid:12)(cid:12) S (cid:16) sD (cid:17)(cid:12)(cid:12)(cid:12) . (9)Using Theorem 3.1, the last line is bounded by X D =1 monic D | R for some R ∈T X s mod D ( s,D )=1 (cid:12)(cid:12)(cid:12) S (cid:16) sD (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:0) q N +1 + ♯ ( S ) · q Q − (cid:1) · X g ∈ F q [ t ] | a g | , (10)where S := { D = 1 monic : D | R for some R ∈ T } . We note that ♯ ( S ) ≤ Q ♯ ( T ) = 2 Q A K ( Q ) . (11)Combining (9), (10) and (11), we obtain (cid:18) X R ∈ F q [ t ] monicdeg R ≤ Q g ′ ( R ) (cid:19) · (cid:12)(cid:12)(cid:12)(cid:12) X g ∈ F q [ t ] a g (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:0) q N +1 + A K ( Q ) · Q q Q − (cid:1) X g ∈ F q [ t ] | a g | . (12) IVISIBILITY PROBLEMS FOR FUNCTION FIELDS 9
Now we set a g = ( g ∈ N ∗ , w ( P ) ≤ w ′ ( P ) for all P ∈ P and hence g ′ ( R ) ≥ g ( R ) for all R ∈ F q [ t ], and X a g ∈ F q [ t ] = X g ∈ F q [ t ] | a g | = N ∗ . From this and (12), we deduce the desired inequality (1). (cid:3) Proof of Theorem 1.4
Our proof depends on Theorem 3.3. We start with the following observation.If
P, R ∈ S and deg
P < deg R then we have U
6≡ − R mod P for all U ∈ S withdeg U > deg P (this follows directly from the definition of S to be a P -set). Fromthis, we conclude that for every P ∈ S there exist at least 1 + [ q deg P /
2] residueclasses R ( P ) , R ( P ) , ..., R w ( P ) ( P ) mod P which do not contain any element of S greater than deg P .Now we use the following sieve. Let Q < N be positive integers. Set N := { f ∈ S : Q < deg f ≤ N } , P := { P ∈ S : deg P ≤ Q } , Ω P : = w ( P ) [ i =1 R i ( P ) if P ∈ P , N ∗ := { f ∈ N : f / ∈ Ω P for all P ∈ P } . We note that N = N ∗ in this case, by definition of S . It follows that A S ( N ) ≤ ♯ ( N ∗ ) + q Q (13)and w ( P ) ≥ q deg P / P ∈ P . Hence, in the notations of Theorem 3.3, we have g ( R ) ≥ R ∈ K (14)and g ( R ) = 0 if R
6∈ K . (15)Using (1), (13), (14) and (15), we obtain A S ( N ) ≤ q N +1 + A K ( Q ) · Q q Q − A K ( Q ) + q Q . Choosing N : = Q + ⌈ log q A K ( Q ) ⌉ , we deduce that A S (cid:0) Q + log q A K ( Q ) (cid:1) ≪ (2 q ) Q . Assume that 0 < β < A S ( N ) ≥ q Nβ for all N large enough. Then itfollows that(2 q ) Q ≫ A S (cid:0) Q + log q A K ( Q ) (cid:1) ≥ A S (cid:0) Q + log q A S ( Q ) (cid:1) ≥ (cid:0) q Q · A S ( Q ) (cid:1) β ≥ q Q ( β + β ) for all Q large enough, which implies β + β ≤ β ≤ / Φ, where Φ = ( √ / A S ( N ) ≤ q (1 / Φ+ ε ) N for infinitely many integers N . (cid:3) Proof of Theorem 1.5
Again, our proof depends on Theorem 3.3. We start with the following ob-servation similar to that at the beginning of the last section. If f, f ′ ∈ F and P is an irreducible monic polynomial, then f + f ′ P . From this, weconclude that for every such P there exist at least 1 + h q deg P / i residue classes R ( P ) , R ( P ) , ..., R w ( P ) ( P ) mod P which do not contain any element of F .Now we use the following sieve. Let Q < N be positive integers. Set N := { f ∈ F : Q < deg f ≤ N } , P := { P : P irreducible and monic and deg P ≤ Q } , Ω P : = w ( P ) [ i =1 R i (cid:0) P (cid:1) if P ∈ P , N ∗ := { f ∈ N : f / ∈ Ω P for all P ∈ P } . By the observation at the beginning of this section, we have ♯ ( F ) ≤ ♯ ( N ∗ ) + q Q (16)and w (cid:0) P (cid:1) ≥ h q deg P / i if P ∈ P . Hence, in the notation of Theorem 3.3, we have g ( R ) ≥ R ∈ K (17)and g ( R ) = 0 if R
6∈ K . (18)Using (1), (16), (17) and (18), we obtain ♯ ( F ) ≤ q N +1 + A K ( Q ) · Q q Q − A K ( Q ) + q Q . IVISIBILITY PROBLEMS FOR FUNCTION FIELDS 11
By the prime number theorem for function fields, we have A K ( Q ) ≥ ♯ { P ∈ F q [ t ] : P monic and irreducible and deg P ≤ Q/ } ≫ q Q/ Q .
Choosing Q := ⌈ N/ ⌉ , we deduce that ♯ ( F ) ≪ q N (2 / ε ) , as claimed. (cid:3) Proof of Theorem 1.6
Proof.
Assume that, contrary to the statement of Theorem 1.6, we havemin { ♯ ( F ) , ♯ ( G ) } > q N (2 / ε ) (19)for N sufficiently large, but f g + 1 is square free for all f ∈ F , g ∈ G . Then, bythe orthogonality relations for multiplicative characters, we have0 = ♯ (cid:8) ( f, g ) ∈ F × G : f g + 1 ≡ P (cid:9) = 1 φ ( P ) X χ mod P χ ( − X f ∈F X g ∈G χ ( f g )for every irreducible monic polynomial P . This implies ♯ { ( f, g ) ∈ F × G : ( f g, P ) = 1 } = χ ( − X f ∈F X g ∈G χ ( f g )= − X χ mod P χ = χ χ ( − X f ∈F X g ∈G χ ( f g ) , where χ is the principal character mod P . Hence, ♯ { ( f, g ) ∈ F × G : ( f g, P ) = 1 } ≤ X χ mod P χ = χ (cid:12)(cid:12)(cid:12) X f ∈F χ ( f ) (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) X g ∈G χ ( g ) (cid:12)(cid:12)(cid:12) . (20)Now we set Q = ⌈ N/ ⌉ and S := { P ∈ F q [ t ] : P irreducible and monic and deg P = Q } . Summing (20) over all irreducible monic polynomials P ∈ S , we get S := X P ∈ S ♯ { ( f, g ) ∈ F × G : ( f g, P ) = 1 }≤ X P ∈ S X χ mod P χ = χ (cid:12)(cid:12)(cid:12) X f ∈F χ ( f ) (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) X g ∈G χ ( g ) (cid:12)(cid:12)(cid:12) . Using the Cauchy-Schwarz inequality, it follows that S ≤ S F S G , (21) where S F = X P ∈ S X χ mod P χ = χ (cid:12)(cid:12)(cid:12) X f ∈F χ ( f ) (cid:12)(cid:12)(cid:12) (22)and S G is defined similarly with G in place of F . On the other hand, using (19),our choice Q = ⌈ N/ ⌉ and ♯ ( S ) ≫ q Q Q , (23)which is a consequence of the prime number theorem for function fields, we have S ≥ X P ∈ S ( ♯ { ( f, g ) ∈ F × G} − ♯ { ( f, g ) ∈ F × G : P | f }− ♯ { ( f, g ) ∈ F × G : P | g } ) ≥ (cid:0) ♯ ( F ) · ♯ ( G ) − ( ♯ ( F ) + ♯ ( G )) q N − Q (cid:1) · ♯ ( S ) ≫ ♯ ( F ) · ♯ ( G ) · q N (1 / − ε/ , (24)where the last line arrives by using (19). Further, we may rewrite the sum S F defined in (22) in the form S F = X P ∈ S (cid:18) X χ mod P χ primitive (cid:12)(cid:12)(cid:12) X f ∈F χ ( f ) (cid:12)(cid:12)(cid:12) + X χ mod Pχ primitive (cid:12)(cid:12)(cid:12) X f ∈F χ ( f ) (cid:12)(cid:12)(cid:12) (cid:19) . Using Theorem 3.2, Q = ⌈ N/ ⌉ and (23), we estimate the right-hand side toobtain S F ≤ (cid:0) q N +1 + 2 ( ♯S ) q Q − (cid:1) · ♯ ( F ) ≪ q N · ♯ ( F ) , (25)and in the same way we get S G ≪ q N · ♯ ( G ) . (26)It follows from (21), (24), (25) and (26) that ♯ ( F ) · ♯ ( G ) · q N (1 / − ε/ ≪ S ≪ q N · ( ♯ ( F ) · ♯ ( G )) / , which implies thatmin { ♯ ( F ) , ♯ ( G ) } ≤ ( ♯ ( F ) · ♯ ( G )) / ≪ q N (2 / ε/ , contradicting (19). (cid:3) References [1] Baier, S.,
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Stephan Baier, Ramakrishna Mission Vivekananda Educational Research In-stitute, Department of Mathematics, G. T. Road, PO Belur Math, Howrah,West Bengal 711202, India; email: email − [email protected] Bansal, Jawaharlal Nehru University, School of Physical Sciences,New-Delhi 110067, India; email: [email protected] Kumar Singh, Ramakrishna Mission Vivekananda Educational Re-search Institute, G. T. Road, PO Belur Math, Howrah, West Bengal 711202,India; [email protected] r X i v : . [ m a t h . N T ] O c t ERRATUM TO THE PAPER “DIVISIBILITY PROBLEMS FORFUNCTION FIELDS”
STEPHAN BAIER, ARPIT BANSAL, RAJNEESH KUMAR SINGH
Abstract.
In [3], we derived three results in additive combinatorics for func-tion fields. The proofs of these results depended on a recent bound for thelarge sieve with sparse sets of moduli for function fields by the first and third-named authors in [1]. Unfortunately, they discovered an error in this paperand demonstrated in [2] that this result cannot hold in full generality. In thepresent paper, we formulate a plausible conjecture under which the said threeresults in [3] remain true and the method of proof goes through using the samearguments. However, these results are now only conditional and still await afull proof.
In [3], we derived the following three results in additive combinatorics for func-tion fields.
Theorem 0.1.
We call a subset S of F q [ t ] a P -set if it consists of monic poly-nomials and no element of S divides the sum of two ( not necessarily distinct ) elements of S of larger degree. Let S be a P -set of pairwise coprime polynomialsin F q [ t ] and denote by A S ( N ) the number of elements of S of degree not exceeding N . If ε > , then there exist infinitely many positive integers N such that A S ( N ) ≤ q N (1 / Φ+ ε ) , where Φ is the golden ratio, given by Φ := √ . Theorem 0.2. If ε > , N ∈ N is sufficiently large, F is a set of monic polyno-mials in F q [ t ] of degree not exceeding N and f + f ′ is square-free for all f, f ′ ∈ F ,then ♯ ( F ) ≤ q N (2 / ε ) . Theorem 0.3. If ε > , N ∈ N is sufficiently large, F , G are sets of monicpolynomials in F q [ t ] of degree not exceeding N and f g + 1 is square-free for all f ∈ F and g ∈ G , then min { ♯ ( F ) , ♯ ( G ) } ≤ q N (2 / ε ) . Mathematics Subject Classification.
The proof of these results depended on the following recent bound for the largesieve with sparse sets of moduli for function fields by the first and third-namedauthors in [1].
Theorem 0.4.
Let Q and N be positive integers, and S be a set of non-zeromonic polynomials in F q [ t ] of degree not exceeding Q . Then X f ∈ S X r mod f, ( r,f )=1 (cid:12)(cid:12)(cid:12)(cid:12) X g ∈ F q [ t ]deg g ≤ N a g e (cid:16) g · rf (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:0) q N +1 + ( ♯S ) · q Q − (cid:1) · X g ∈ F q [ t ]deg g ≤ N | a g | , (1) where a g are arbitrary complex numbers. For a generic set S , this is what one would expect, but unfortunately, the firstand third-named authors discovered an error in this paper [1] and demonstratedin [2] that this result cannot hold in full generality. The counterexample providedthere shows that one needs an additional factor of significant size. However, asfar as this counterexample is concerned, this factor is bounded by q ε ( Q + N ) . Wetherefore propose the following plausible conjecture which is a slight weakening of(1) and does not contradict the counterexample given in [2]. Except for particularcases, it seems to be difficult to establish this conjecture, though, and it is notclear if it holds in full generality or if some additional conditions on the set S arerequired. Conjecture 0.5.
Let ε > be given. Let Q and N be positive integers and S bea set of non-zero monic polynomials in F q [ t ] of degree not exceeding Q . Then X f ∈ S X r mod f, ( r,f )=1 (cid:12)(cid:12)(cid:12)(cid:12) X g ∈ F q [ t ]deg g ≤ N a g e (cid:16) g · rf (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≪ q,ε q ε ( Q + N ) (cid:0) q N + ( ♯S ) · q Q (cid:1) · X g ∈ F q [ t ]deg g ≤ N | a g | , where a g are arbitrary complex numbers. Under this conjecture the Theorems 1,2,3 stated above remain true, and themethods of proof are unaffected. However, these theorems are now only condi-tional and still await a full proof.
References [1] Baier, Stephan; Singh, Rajneesh Kumar;
Large sieve inequality with power moduli forfunction fields , J. Number Theory 196 (2019) 1–13.[2] Baier, Stephan; Singh, Rajneesh Kumar;
The large sieve with square moduli in functionfields , arXiv:1910.05043.[3] Baier, S.; Bansal, A.; Singh, R. Kumar,
Divisibility problems for function fields , ActaMath. Hungar. 156 (2018), no. 2, 435–448.