The Erdős–Moser Sum-free Set Problem

We show that there is an absolute $c>0$ such that if $A$ is a finite set of integers, then there is a set $S\subset A$ of size at least $\log ^{1+c}|A|$ such that the restricted sumset $\{s+s^{\prime }:s,s^{\prime }\in S\text{ and }s\neq s^{\prime }\}$ is disjoint from $A$ . (The logarithm here is to base $3$ .)

On the size of a restricted sumset with application to the binary expansion of √d

For any A ? N, let U(A,N) be the number of its elements not exceeding N. Suppose that A + A has V (A,N) elements not exceeding N, where the elements in the sumset A + A are counted with multiplicities. We first prove a sharp inequality between the size of U(A,N) and that of V (A,N) which, for the upper limits ?(A) = lim supN?? U(A,N)N-1/2 and ? (A) = lim sup N?? V (A,N)N-1, implies ?(A)2 ? 4 ? (A)/?. Then, as an application, we show that, for any square-free integer d > 1 and any ? > 0, there are infinitely many positive integers N such that at least (?8/ ?- ?) ?N digits among the first N digits of the binary expansion of ?d are equal to 1.

A robust version of Freiman's 3k–4 Theorem and applications

We prove a robust version of Freiman's 3k – 4 theorem on the restricted sumset A+ΓB, which applies when the doubling constant is at most (3+$\sqrt{5}$)/2 in general and at most 3 in the special case when A = −B. As applications, we derive robust results with other types of assumptions on popular sums, and structure theorems for sets satisfying almost equalities in discrete and continuous versions of the Riesz–Sobolev inequality.