On large subsets of $\mathbb{F}_q^n$ with no three-term arithmetic progression

Abstract

In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of F n q with no three terms in arithmetic progression by c n with c < q. For q = 3, the problem of finding the largest subset of F n 3 with no three terms in arithmetic progression is called the cap problem. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz [BK12], was on order n −1−ǫ 3 n .The problem of finding large subsets of an abelian group G with no three-term arithmetic progression, or of finding upper bounds for the size of such a subset, has a long history in number theory. The most intense attention has centered on the cases where G is a cyclic group Z/NZ or a vector space (Z/3Z) n , which are in some sense the extreme situations. We denote by r 3 (G) the maximal size of a subset of G with no three-term arithmetic progression. The fact that r) was first proved by Brown and Buhler [BB82], which was improved to O(3 n /n) by Meshulam [Mes95]. The best known upper bound, O(3 n /n 1+ǫ ), is due to Bateman and Katz [BK12]. The best lower bound, by contrast, is around 2.2 n [Ede04].The problem of arithmetic progressions in (Z/3Z) n has sometimes been seen as a model for the corresponding problem in Z/NZ. We know (for instance, by a construction of Behrend [Beh46]) that r 3 (Z/NZ) grows more quickly than N 1−ǫ for every ǫ > 0. Thus it is natural to ask whether r 3 ((Z/3Z) n ) grows more quickly than (3 − ǫ) n for every ǫ > 0. In general, there has been no consensus on what the answer to this question should be.In the present paper we settle the question, proving that for all odd primes p, r 3 ((Z/pZ) n ) 1/n is bounded away from p as n grows.The main tool used here is the polynomial method, in particular the use of the polynomial method developed in the breakthrough paper of Croot, Lev, and Pach [CLP16], which drastically improved the best known upper bounds for r 3 ((Z/4Z) n ). In this case, they show that a subset of G with no three-term arithmetic progression has size at most c n for some c < 4. In the present paper, we show that the ideas of their paper can be extended to vector spaces over a general finite field.

BME VIK Annual Research Report on Electrical Engineering and Computer Science 2016

“…• Ellenberg and Gijswijt [171] by a modification of the argument managed to give an exponentially small upper bound for r3(Z3n), which question is known as the "cap set problem". Namely, they proved that r3(Z3n)=O(2.756n).…”

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“…Subsequently, Bateman and Katz [7] improved the bound to |A| ≤ (c/n 1+ε ) · 3 n for some small unspecified constant ε > 0. Very recently, breakthrough works of Croot, Lev and Pach [23] and Ellenberg and Gijswijt [27] improved the bound to c n for c < 3 an absolute constant. Their techniques are quite different; we refer the interested reader to the papers for details.…”

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“…First, Croot, Lev, and Pach obtained an upper bound of this exponential type for subsets of Z n 4 with no 3-term arithmetic progression [12]. Then, barely a week later, Jordan Ellenberg and Dion Gijswijt independently saw how to modify the argument of Croot, Lev, and Pach to give a similar bound for the cap-set problem itself, thereby giving a positive answer to the question above [13].…”