A Combinatorial Proof of the Existence of Dense subsets in R without the "Steinhaus" like Property

On the uniform distribution in residue classes of dense sets of integers with distinct sums

A set ${\cal A} \subseteq \Set{1,...,N}$ is of type $B_2$ if all sums $a+b$, with $a\ge b$, $a,b\in {\cal A}$, are distinct. It is well known that the largest such set is of size asymptotic to $N^{1/2}$. For a $B_2$ set ${\cal A}$ of this size we show that, under mild assumptions on the size of the modulus $m$ and on the difference $N^{1/2}-\Abs{\cal A}$ (these quantities should not be too large) the elements of ${\cal A}$ are uniformly distributed in the residue classes mod $m$. Quantitative estimates on how uniform the distribution is are also provided. This generalizes recent results of Lin…