DIFFERENCE SETS AND THE IRREDUCIBLES IN FUNCTION FIELDS. 1. Introduction

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1 DIFFERENCE SETS AND THE IRREDUCIBLES IN FUNCTION FIELDS Abstract. Let A be a subset of the polnomials of degree less than N over a finite field F q. Let r be an non-zero element of F q. We show that if the difference set A A does not contain elements of the form P + r, where P is a monic, irreducible polnomial, then A Cq N c log N N, where C and c are constants depending onl on q.. Introduction In a series of papers, Sárköz [9, 0, ] investigated the set of differences of a set of positive densit in the integers. He proved the following theorem in []: Theorem. If A is a subset of positive densit of the integers, then there exist two distinct elements a, a of A such that a a = p for some prime p. Actuall, he showed that if A {,..., N} is such that the difference set A A does not contain elements of the form p, where p is prime, then A N log 3 N 3 log 4 N log 2 N 2, where log i N denotes i iterations of the log function. To date, the current record upper bound on A is due to Ruzsa and Sanders, who showed in [8] that: A N exp c 4 log N for some constant c > 0. In view of the man analogies between the integers and the ring F q [t] of polnomials over a finite field F q, it is natural to ask for the analog of Sárköz s theorem in the setting of F q [t]. Let us fix a field F q of q elements. Let G N be the set of all polnomials in F q [t] of degree less than N. In this paper, we prove the following: Theorem 2. Let r be a fixed element of F q. Let A be a subset of size δq N of G N such that the difference set A A does not contain elements of the form P + r, where P is a monic, irreducible polnomial. Then we have δ Cq c N log N for some constants C and c depending onl on q. Date: August 3, Mathematics Subject Classification. P55, T55, L07. Ke words and phrases. Difference sets, irreducible polnomials, function fields, circle method. The research of the second author is supported in part b NSF Grant DMS and NSA Young Investigators Grant H

2 2 Note that in the F q [t] setting, q N plas the role of N in the integer setting. Thus, Ruzsa and Sanders s bound corresponds to a bound on δ of the form Cq c 4 N in the F q [t] case, which is weaker than our bound. It is possible to adapt Kamae and Mendès France s [3] or Furstenberg s [, 2] approaches to Theorem in the F q [t] setting, but these methods are not quantitative i.e., not giving explicit dependence of δ on N. Our arguments run in parallel with Ruzsa and Sanders s approach. We are able to obtain better bounds than in the integer case due to better error terms in the exponential sum estimates, which in turn comes from the validit of the Generalized Riemann Hpothesis for F q [t]. In another direction, Ruzsa constructed in [7] an example of a subset of {,..., N} whose difference set does not contain elements of the form p where p is prime, and whose size is exp log o log N log log N. A straightforward adaptation of Ruzsa s construction gives the following lower bound of the same tpe in the F q [t] setting, of which we will omit the proof: log q qq +o log N Proposition 3. For all integers N >, there is a set A G N such that A q and A A does not contain P + x, for all not necessaril monic irreducible polnomial P and all x F q. If we merel require P to be monic, then we can do much better if q > 3. Indeed, if q > 3, then let A be the set of all polnomials in G N whose coefficients are in R, where R is a subset of F q whose difference set does not contain. Then, all the differences in A are not monic, and A is of size q cn. It ma be interesting for one to stud the maximal size of a subset of G N whose difference set avoids the monic polnomials. It should be mentioned that the conclusions of Theorem remain true if we replace {p : p prime} b the set of the squares. We plan to treat the F q [t] analog for the squares and more generall, k-th powers in a future paper. Acknowledgments. We would like to thank Tom Sanders and Terence Tao for helpful discussions. This research was started while the second author was a member of the Institute for Advanced Stud, and he would like to thank the School of Mathematics for their hospitalit. N 2. Notation and preliminaries 2.. Notation. For k N, let fk and gk be functions of k. If gk is positive and there exists a constant c > 0 such that fk cgk for all k N, we write fk gk. In this paper, all the implicit constants depend onl on q. The value of r is also fixed throughout the paper. Let K = F q t be the field of fractions of F q [t], and let K = F q /t be the completion of K at. Each element α K ma be written in the form α = i w a it i for some w Z and a i = a i α F q i w. If a w 0, we sa that ord α = w, and we write α for q ord α. We adopt the conventions that ord 0 = and 0 = 0. We write α for i min{w, } a it i. Also, it is often convenient to refer to a as being the residue of α, denoted b res α. For a real number N, we let N denote q N. Hence, if x is a polnomial in F q [t], then x < N if and onl if the degree of x is strictl less than N. Recall that G N = { x F q [t] x < N }, and let P N denote the set of all monic, irreducible polnomials in G N.

3 DIFFERENCE SETS AND THE IRREDUCIBLES IN FUNCTION FIELDS 3 We define the F q [t]-analogue of the von-mangoldt function Λ : F q [t] Z b { ord ϖ, if ϖ k = x, where ϖ is a monic, irreducible polnomial, Λx = 0, otherwise. Let us also define λ m = λ m ; N = { ord m + r, if m + r is monic, irreducible, and of degree less than N, 0, otherwise. For a polnomial g, let φg denote the Euler totient function of g, i.e. the number of units in the ring F q [t]/m. It is eas to see that for ever ɛ > 0, φg ɛ g ɛ The circle method in F q [t]. Consider the compact additive subgroup T of K defined b T = { α K : α < }. Given a Haar measure dα on K, we normalize it so that T dα =. Thus, if N is the subset of K defined b N = { α K ord α < N }, where N is an integer, then the measure of N, mesn = α N dα, is equal to N. We are now able to define the exponential function on F q [t]. Suppose that the characteristic of F q is p. Let ez denote e 2πiz, and let tr : F q F p denote the familiar trace map. There is a non-trivial additive character e q : F q C defined for each a F q b taking e q a = etra/p. This character induces a map e : K C b defining, for each element α K, the value of eα to be e q res α. The orthogonalit relation underling the Fourier analsis of F q [t], established in [4, Lemma ], takes the shape {, if h = 0, ehα dα = 0, if h F q [t] \ {0}. T For a function f defined on F q [t], of compact support, let us define its Fourier transform b fα = eαxfx. For a monic polnomial m F q [t], we define the weighted exponential sum over the irreducible polnomials in arithmetic progression b h m α = h m α; N = λ m ; Neα. F q[t] Thus for an set A G N, b, we have A α 2 h m α dα = T x A x 2 A F q[t], x x 2 = λ m. For a, g F q [t] with a < g, where g is a monic polnomial, we write M a,g,η = { α T α a/g < η }.

4 4 We also write and M g,η = M g,η = a < g a < g a,g= M a,g,η M a,g,η. 3. Major and minor arc estimates for h m α In this section, we obtain the necessar major and minor arc estimates for h m α that are needed in the proof of Theorem 2. Before doing this, we will need to establish additional notation. Let P R denote the set of monic irreducible polnomials of ord R, and let S R denote the set of monic polnomials of ord R. For β T, let τ R β = τβ; R = x S R eβ. 3.. Minor arc estimates. We now recall [5, Lemma 23], which will be used to derive our minor arc estimate. Lemma 4. Let m F q [t] be a monic polnomial, and let b F q [t] with b < m and b, m =. Let a, g F q [t] with, a < g, and a, g =. Suppose that m q 2 N 2/5 N, g q N m, and α M a,g, g 2. Then, we have Λeα N 4/5 m N 4 + g N 3 + NN 9/2 m /2 g /2 + N /2 N 9/2 m /2 g /2. G N b mod m Lemma 5. Let m F q [t] be a monic polnomial with m q 2 N 2/5 N, and let Q be a positive real number. Let a, g F q [t] with, a < g Q q N m, and a, g =. Suppose that α M a,g, Q g. Then, we have h m α; N N 4/5 m N 4 + Q m N 3 + NN 9/2 m /2 g /2 + N /2 N 9/2 m Q /2. Proof. We have h m α; N = ord m + r eα G N ord m m+r P N = e αr/m Λzeαz/m + O z G N z r mod m z G N z r mod m ϖ monic irred ϖ 2 < N ord ϖ + Λzeαz/m + N /2 N. ϖ monic irred ϖ 3 < N ord ϖ + 2

5 DIFFERENCE SETS AND THE IRREDUCIBLES IN FUNCTION FIELDS 5 B the Dirichlet approximation theorem, there exist a, g F q [t] with g monic, a, g =, g m Q, and α/m a /g < m Q g. Then, a g a { a max mg g α α, m m From the above inequalit, we ma deduce that a } < max{ m mg Q g, m Q g }. a mg ag < max{ Q g, Q g }. Suppose for the moment that g < g Q. We then have a mg ag <, impling that a mg = ag. Since g, a =, it follows that g g, and we ma deduce that g g, which provides a contradiction. Hence, we have g g. Appling Lemma 4 with the approximation a /g to α/m, we find that Λzeαz/m N 4/5 m N 4 + g N 3 + NN 9/2 m /2 g /2 + N /2 N 9/2 m /2 g /2 z G N z r mod m N 4/5 m N 4 + Q m N 3 + NN 9/2 m /2 g /2 + N /2 N 9/2 m Q /2. 3 The lemma now follows b combining 2 and Major arc estimates. Lemma 6. Let m F q [t] be a monic polnomial, and let r F q [t] with r < m and r, m =. Let Q and Q 2 be positive real numbers. Let a, g F q [t] with, a < g Q 2, and a, g =. Let α M a,g, g. Let M N with M Q Q 2. Then, we have µg m e ar m g φmφgm+ord m τ Mα a/g + O M 3/2 m /2 M+ord m 2, if m, g =, min M/ Q2, eα = Q S M O M 3/2 m /2 M+ord m 2, otherwise. m+r P M+ord m min M/ Q2, Q Here, m denotes the multiplicative inverse of m modulo g. Proof. This lemma follows from the proofs of [5, Lemmas 7-] upon redefining L to be min[m Q 2 ], [Q ] and M a,g to be M a,g, g. Q Lemma 7. Let m F q [t] be a monic polnomial, and r F q [t] with r < m and r, m =. Let Q and Q 2 be positive real numbers. Let a, g F q [t] with, a < g Q 2, and a, g =. Let α M a,g, g. Then, we have Q µg m e ar m g φmφg e α a/gx + O fn, Q, Q 2 ; m, if m, g =, h m α; N = x G N ord m x monic O fn, Q, Q 2 ; m, otherwise, where fn, Q, Q 2 ; m = N /2 N 3 Q2 + N 3/2 N 3 m Q.

6 6 Proof. Note that h m α; N = ord m + r eα = = G N ord m m+r P N N ord m X=0 N ord m X= Q 2 X + ord m X + ord m B appling Lemma 6, we deduce that µg m e ar m g φmφg h m α; N = O fn, Q, Q 2 ; m, x G N ord m x monic eαz z S X mz+r P X+ord m eαz z S X mz+r P X+ord m + O + O Q2 m φm. e α a/gx + O fn, Q, Q 2 ; m, if m, g =, where fn, Q, Q 2 ; m = N /2 N 2 Q2 + N 3/2 N 2 m Q. This completes the proof of the lemma. otherwise, 4. Energ Increments In this section, we will establish lemmas concerning energ increments. These are analogous to those found in [8, Section 7]. Lemma 8. Suppose that L N and m, r F q [t] with m 0. Also, suppose that A G N with A = δ N. Let B = {ml + r l < L}. Furthermore, suppose that 2x A δ GN B cδ 2 N L2. Then, there exists x F q [t] such that A B x + cδ L + O N m L 2. Proof. Note that A B GN B x = A x GN B B x = δ N L 2 + Oδ m L 3. Also, we have 2x GN B = N L2 + O m L

7 DIFFERENCE SETS AND THE IRREDUCIBLES IN FUNCTION FIELDS 7 Since A δ GN B 2x cδ 2 N L2, we ma deduce from 4 and 5 that 2x A B cδ 2 N L2 + 2δ A B GN B x δ 2 GN B 2x = cδ 2 N L2 + 2δ 2 N L2 δ 2 N L2 + Oδ m L 3 = + cδ 2 N L2 + Oδ m L 3. B the triangle inequalit, we have 2x A B sup A B x x F q[t] The lemma now follows b combining 6 and 7. 6 A B x = δ N L sup A B x. 7 x F q[t] Lemma 9. Suppose that η > 0, N N, and g F q [t] \ {0}. Suppose that A G N with A = δ N. Write E A,g,η = δ 2 N α A δ GN 2 dα. M g,η Then, there exist L N with L g min{η, E A,g,η A } and r F q [t] such that for B = {lg + r l < L}, we have A B δ + E A,g,η /2 L. Proof. Let L N be a parameter to be chosen later with L η g, and let D = {gl l < L}. For x D and α M a,g,η, we have α a/gx < η g q L q, which implies that eαx = eax/g. Hence, for α M a,g,η, we have D α = eαx = x D x D eax/g = l < L eal = L. It follows from the triangle inequalit that δ 2 N L2 E A,g,η α A δ GN 2 D α 2 dα M g,η α A δ GN 2 D α 2 dα T = 2x. A δ GN D B Lemma 8, there exists x F q [t] such that A D x + E A,g,η δ L + O N g L 2. Thus, there is a choice of L N for which L g min η, E A,g,η A and A D x δ + E A,g,η L. 2 The lemma now follows.

8 8 Lemma 0. Suppose that η > 0, N N, and g F q [t] \ {0}. Suppose that A G N with A = δ N. Write EA,g,η = δ 2 N α A δ GN 2 dα. Suppose that for K N, we have g K M g,η φg E A,g,η c. Then, there exist g, r F q [t] with g K and L N with L g min{η, δc N} such that for B = {lg + r l < L}, we have A B δ + c c L, where c = c q is a positive constant depending at most on q. Proof. Define E A,g,η as in Lemma 9, and write I A,a,g,η = δ 2 N Note that Also, we have Therefore, we have g K g φg E A,g,η = h K/ g h monic g K b the hpothesis. Also, we have g φg = g K g K d g d monic M a,g,η A δ GN α 2 dα. g K = g K = g K = g K g monic g h φg h g φg g φg E A,g,η K µd 2 φd = d K d monic g φg g φg g φg E A,g,η r < g I A,r,g,η g h=g r < g g monic r,g = g h=g g monic E A,g,η h K/ g h monic h K/ g h monic g K g monic µd 2 φd I A,r h,g h,η g h φg h. K φg. φg E A,g,η Kc 8 g K/ d K d K d monic d φd K. 9

9 DIFFERENCE SETS AND THE IRREDUCIBLES IN FUNCTION FIELDS 9 B 8 and 9, there exists a monic polnomial g with g K and E A,g,η c. The result now follows from Lemma An iteration and the proof of Theorem 2 In this section, we first prove the following lemma. Lemma. Suppose that N C 0, A G N, A = δ N, δ N κ, and m N κ, where κ is a small, absolute constant. If the difference set A A does not contain elements of the form P +r m, where P is monic, irreducible, then there exists an integer N, a set A G N of densit δ in G N, and a monic polnomial m such that the following hold: δ + C δ, 2 m C 2 δ 2 m, 3 N C 3 δ N m 9 N, 4 A A { P +r m : P monic, irreducible} =, where C 0, C, C 2, and C 3 are constants depending at most on q. Proof. Let f A = A δ GN, the balanced function of A. Note that T f A α 2 dα = δ δ 2 N. Let us consider the expression I = f A α 2 h m α; Ndα. T Let us first compute I explicitl. We have I = Ax + Axλ m + δ 2 G N x + G N xλ m x A, G N x G N, G N δ G N x + Axλ m δ G N xax + λ m x G N, G N x G N, G N = δ 2 N λ m, since b hpothesis the first term is zero. N Let Q = cδ 2, Q2 = c δ 2 N 9 m 3, where c is a sufficientl small constant to be chosen N 9 m 4 later. B the Dirichlet approximation theorem, the sets M a,g, g Q, where a, g =, g is monic, and a < g Q, are disjoint and form a partition of T. Let us define the major arcs M = M a,g, g Q and the minor arcs m = a < g, a,g= g Q 2, a < g, a,g= Q 2 < g Q M a,g, g Q 0

10 0 so that T = M m. Claim. The contribution from the minor arcs in I is small. Indeed, b Lemma 5 we have sup h m α; N N 4/5 m N 4 + Q m N 3 + NN 9/2 m /2 /2 Q2 + N /2 N 9/2 /2 m Q. α m B our choices of Q and Q 2, all the four terms are dominated b δ N m. Thus, f A α 2 h m α; Ndα δ N f A α 2 dα δ N 2 2 m m. m B Lemma 7, we see that λ m N 2 φm N 2 m if m N κ and N is sufficientl large. Thus, for an appropriate choice of c, we have that for all sufficientl large N, f A α 2 h m α, Ndα 2 δ2 N λ m. m Now let Q 3 = c δ 2, where c is a sufficientl large positive constant to be chosen later, and let us split the major arcs M into two parts, M = M M2, where M = M a,g, g Q and M 2 = a < g, a,g= g Q 3, a < g, a,g= Q 3 < g Q 2 T M a,g, g Q. Claim 2. The contribution from M 2 in I is small. Indeed, for α M a,g, g Q, where Q 3 < g Q 2, Lemma 5 implies that Therefore, h m α; N N φgφm f A α 2 h m α, Ndα δ M 2 N g /2 φm T f A α 2 dα N Q /2 3 φm δ λ m. λ m δ 2 N λ m. Thus, for an appropriate choice of c, we have that for all sufficientl large N, f A α 2 h m α, Ndα M 2 4 δ2 N λ m. Therefore, for sufficientl large values of N, we have f A α 2 h m α; Ndα M 4 δ2 N λ m.

11 DIFFERENCE SETS AND THE IRREDUCIBLES IN FUNCTION FIELDS For α M a,g, g Q, where g Q 3, we have h m α; N φg there exists a positive constant c satisfing φg g Q 3 for all sufficientl large values of N. a < g a,g= M a,g, g Q f A α 2 dα cδ 2 N λ m. We conclude that We can now appl Lemma 0 for K = Q 3 and η = Q, with the observation that M M a,g, g Q g,η. a < g a,g= We can thus find g, s F q [t] with g Q 3 = c δ 2 and L with L g min Q, cδ N N δ 2 Q = cδ 4 N 9 m 4 such that A {gl + s : l < L} δ + c c L, where c is the constant in Lemma 0. Let us now set N = L, A = {l : l < L, gl + s A}, and m = gm. Clearl, if A A avoids { P +r m : P monic, irreducible}, then A A avoids { P +r m : P monic, irreducible}. Proof of Theorem 2. Suppose for a contradiction, A A does not contain elements of the form P +r, where P is a monic, irreducible polnomial. We use Lemma to successivel construct a sequence of quadruples N i, A i, δ i, m i such that A i G Ni, A i = δ i Ni, and the following hold for ever i: N 0, A 0, δ 0, m 0 = N, A, δ,, 2 δ i+ + C δ i, 3 m i+ C 2 δ 2 i m i, 4 N i+ C 3 δi N i m i 9 Ni, 5 A i A i { P +r m i : P monic, irreducible} =. We claim that if N is sufficientl large depending on δ, we can construct a sequence of Z = C 4 log δ + quadruples, where C 4 is a sufficientl large constant to be chosen later, at which point we have a contradiction since δ Z >. Once we have N i, A i, δ i, m i, we can produce N i+, A i+, δ i+, m i+ as long as the hpothesis of Lemma is satisfied, i.e. Ni C 0, δ N i κ, and m i N i κ. Since the sequence N i is decreasing and the sequence m i is increasing, it suffices to show that for N sufficientl large, for an sequence of triples N i, δ i, m i Z i=0 satisfing the recursive relations,2,3 above, we have N Z C 0, m Z N κ, δ N κ. B induction it is eas to see that for ever i {0,..., Z},

12 2 δ i + C i δ, m i C i 2 + C ii δ 2i. Therefore, for ever i = {0,..., Z}, N i+ C 3 N 9 Consequentl, + C i δ C i 2 + C ii δ 2i 9 Ni C 3 N 9 C i 2 + C i2 δ 2i+ 9 Ni. N Z N CZ 3 N 9Z C2 Z2 + C Z3 /6 δ Z2 9. Let us verif that the conditions of Lemma hold for i = Z, that is, NZ C 0, δ N Z κ, and m Z N Z κ. Notice that since Z = C 4 log δ +, for an appropriate value of C 4, we have C2 Z2 + C Z3 /6 δ Z2 9 C 2 Z + C ZZ δ 2Z /κ mz /κ. Thus, NZ m Z /κ C3 if N Z N, which holds if N 9Z log N C 5 log/δ +, where C 5 is a large constant. If we choose C 5 large enough, then we also have N Z C 0 and N Z δ /κ as well, thus completing the proof of Theorem 2. We have a few concluding remarks. An inspection of the proof of Lemma shows that, in order to have a densit increment on G N, it suffices to have few solutions to a a 2 = P +r m, where a, a 2 A and P is a monic, irreducible polnomial as opposed to none at all. Incorporating this observation into the iteration, we have a contradiction even if the number of solutions to a a 2 = P + r is small enough as opposed to none at all. Thus, we can actuall give a lower bound for the number of such solutions. Precisel, let R m A = be the number of weighted solutions. Then, we have {a a 2 = : a, a 2 A}λ m ; N N 2 RA = R A Cδ N cδ, where Cδ and cδ are constants depending on q and δ. This bound falls short of the expected order of magnitude N 2. It is possible to use alternative methods to show that the number of weighted solutions is indeed of order N 2 see e.g. [2, Section 0.2]. Obtaining the optimal dependence of Cδ on δ, however, is et another interesting problem. References [] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analse Math , [2] H. Furstenberg, Recurrence in Ergodic Theor and Combinatorial Number Theor, Princeton Univ. Press, 98. [3] T. Kamae & M. Mendès France, Van der Corput s difference theorem, Israel J. Math , [4] R. M. Kubota, Waring s problem for F q[x], Dissertationes Math. Rozpraw Mat , 60pp. [5] Y.-R. Liu & C. V. Spencer, A prime analogue of Roth s theorem in function fields, preprint.

13 DIFFERENCE SETS AND THE IRREDUCIBLES IN FUNCTION FIELDS 3 [6] M. Rosen, Number theor in function fields, Springer-Verlag, [7] I. Z. Ruzsa, On measures on intersectivit, Acta Math. Hungar , [8] I. Z. Ruzsa & T. Sanders, Difference sets and the primes, Acta Arith , [9] A. Sárköz, On difference sets of sequences of integers, I., Acta Math. Acad. Sci. Hungar , [0] A. Sárköz, On difference sets of sequences of integers, II., Ann. Univ. Sci. Budapest. Etvs Sect. Math , [] A. Sárköz On difference sets of sequences of integers, III., Acta Math. Acad. Sci. Hungar , [2] Vaughan, The Hard-Littlewood method, 2nd ed., Cambridge Univ. Press, 997. T. H. Lê, School of Mathematics, Institute for Advanced Stud, Einstein Drive, Princeton, NJ address: leth@math.ias.edu C. V. Spencer, Department of Mathematics, Kansas State Universit, 38 Cardwell Hall, Manhattan, KS address: cvs@math.ksu.edu