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On sets with small additive doubling in product sets

Dmitrii Zhelezov

2015
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Journal of Number Theory
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The present thesis consists of five research papers in the areas of combinatorial number theory and arithmetic combinatorics. Each paper is devoted to a specific realisation of the general intuition that, vaguely speaking, a set cannot be simultaneously additively and multiplicatively structured. Papers I and II study arithmetic progressions of maximal length in product sets. In Paper I it is proved that if B is a set of N positive integers such that B · B contains an arithmetic progression of
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... ength M then N ≥ π(M ) + M 2/3−o(1) . On the other hand, we present examples for which N < π(M ) + M 2/3 . The main tool is a reduction of the original problem to the question of an approximate additive decomposition of the 3-sphere in F n 3 which is the set of 0-1 vectors with exactly three non-zero coordinates. In particular, it is proved that such a set cannot be contained in a sumset A + A unless |A| n 2 . In Paper II the same problem of bounding the maximal length of an arithmetic progression in a product set is considered in the complex setting, that is, elements of the set B are now allowed to be complex numbers. In this case we were able to prove a reasonably strong bound only assuming the Generalised Riemann Hypothesis. The obtained bound is for any positive and some constant C . Paper III explores a similar, but more general question, namely to bound the maximal size of a set with small doubling contained in a product set B · B. A set A is said to have small doubling if the size of the sumset A + A is bounded by K|A|, where K is some absolute constant. It holds for example when A is an arithmetic progression of arbitrary length. Let (A n ), (B n ) be sequences of sets such that uniformly holds |A n + A n |/|A n | ≤ K ii and A n ⊂ B n · B n . Under the condition that the sizes of the elements in B n are polynomialy bounded with repect to |B n |, it is proved that |A n | = o(|B n | 2 ). In particular, it follows that under this condition the additive energy of B n · B n is asymptotically o(|B n | 6 ), which, in turn, gives the classical Erdős Multiplication Table Theorem as a special case. Paper IV is dual to Paper III and gives an upper bound for the size of a set A with small multiplicative doubling contained in a sumset B + B. Using different methods from those of Paper III, in particular, the Subspace theorem, Roche-Newton and the author proved the unconditional bound |A| = O(|B| 2 log −1/3 |B|), which implies that the multiplicative energy of a sumset B + B is bounded from above by |B| 6 exp(−O(log 1/3− |B|)). The bounds are then applied to give a partial result towards an inverse sum-product problem, conjectured in the paper. Paper V deals with sum-product type problems in finite fields. It is proved that for sets A, B, C ⊂ F p with |A| = |B| = |C| ≤ √ p and a fixed 0 = In particular, |A · (A + 1)| |A| 1+1/26 and max(|A · A|, |(A + 1) · (A + 1)|) |A| 1+1/26 . The first estimate improves an earlier bound by Roche-Newton and Jones. In the general case of a field of order q = p m , m ≥ 2 similar estimates are obtained with the exponent 1 + 1/559 + o(1) under the condition that A · B does not have large intersection with any subfield coset, answering a question of Shparlinski. The paper concludes with an estimate for the additive energy of a multiplicative subgroup, which is used to obtain an explicit power-saving bound for Gauss sums over multiplicative subgroups of order at least q 28/57+o(1) . To our knowledge, such a bound is not currently present in the literature since extracting explicit bounds from a more general result of Bourgain and Chang seems to be hard.

doi:10.1016/j.jnt.2015.04.029
fatcat:sdc3b7txlray7f5zeoom5pxbcm