### On sets with small additive doubling in product sets

Dmitrii Zhelezov
2015 Journal of Number Theory
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
more » ... 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.