Discretized sum-product for large sets

Sum and product of different sets

Let A and B be two finite sets of numbers. The sum set and the product set of A,B are A+B : = {a+b : a ∈ A, b ∈ B}, and AB : = {ab : a ∈ A, b ∈ B}. We prove that A + B is as large as possible when AA is not too big. Similarly, AB is large when A + A is not too big. The methods rely on the λp constant of A, bound on the number of factorizations in a generalized progression containing A, and the ...

Quantitative Sum Product Estimates on Different Sets

Let Fp be a finite field of p elements with p prime. In this paper we show that for A,B ⊂ Fp with |B| ≤ |A| < p 1 2 then

LARGE SUM-FREE SETS IN Z/pZ

We show that if p is prime and A is a sum-free subset of Z/pZ with n := |A| > 0.33p, then A is contained in a dilation of the interval [n, p−n] (mod p).

The Sum-product Estimate for Large Subsets of Prime Fields

Let Fp be the field of prime order p. It is known that for any integer N ∈ [1, p] one can construct a subset A ⊂ Fp with |A| = N such that max{|A+ A|, |AA|} p|A|. One of the results of the present paper implies that if A ⊂ Fp with |A| > p2/3, then max{|A+ A|, |AA|} p|A|.

Product constructions for large sets of resolvable MTSs and DTSs