A low-energy decomposition theorem

Abstract

We prove that any finite set of real numbers can be split into two parts, one part being highly non-additive and the other highly nonmultiplicative.Similarly, the multiplicativity of a set is measured by its multiplicative energy E × (A) = card{a ∈ A 4 : a 1 a 2 = a 3 a 4 }.

On the size of the set AA+A

“…Let us fix λ 1 , λ 2 , λ 3 , λ 4 with λ 4 = λ 2 . Note that this assumption implies that λ 4 = λ 1 , λ 2 , λ 3 . Suppose…”

“…The classical tool in additive combinatorics for this situation is the Balog–Szemerédi–Gowers Theorem, which says that if is large then contains a large subset with small sum set. However, recent progress, particularly in [, , ], has led to the development of different tools which are more effective than the Balog–Szemerédi–Gowers Theorem in the sum–product setting. In particular we will use the following result, which is [, Theorem 12].…”

Number Theory and Applications, 2

“…Для доказательства нижних оценок в теоремах 3, 4 мы используем небольшую моди-фикацию конструкции из [5]. Контрпример представляет собой так называемое ( + Λ)-множество, см.…”

Analogues of the Balog–Wooley Decomposition for Subsets of Finite Fields and Character Sums with Convolutions

“…For comparison, note that the decomposition results in [2] and [19] are applicable below this range. This is because the main tool in [2] and [19] is a strong new point-plane incidence bound of Rudnev [18], whereas our main tool is the Weil bound. It may be within reach to obtain a version of Theorem 1.1 which is non-trivial for smaller sets by finding a way to apply new results in incidence theory, but we have been unable to do this in the present paper.…”

“…Given three sets A, B, C Ď F q we define the following sums of additive and multiplicative characters Our interest to these sums is motivated by recent progress in bounds of additive and multiplicative character sums involving three sets, see [6,15] and [2,11,20], respectively.…”