Finite addition theorems. I (Q1121319)

The author proves that, given natural numbers N, k and a set \(A\subseteq \{1,2,...,N\}\) with \(| A| \geq (N/k)+1\), there exist integers d,\(\ell,m\); \(1\leq d<k\); \(1\leq \ell <118k\), such that \(\ell A\) contains the arithmetic progression \(\{(m+i)d|\) \(i=1,...,N\}\). A second theorem shows that, roughly speaking, the factor 118 cannot be replaced by 1.