ALMOST ALL PRIMES HAVE A MULTIPLE OF SMALL HAMMING WEIGHT

Abstract

We improve recent results of Bourgain and Shparlinski to show that, for almost all primes $p$, there is a multiple $mp$ that can be written in binary as $$\begin{eqnarray}mp=1+2^{m_{1}}+\cdots +2^{m_{k}},\quad 1\leq m_{1}<\cdots <m_{k},\end{eqnarray}$$ with $k=6$ (corresponding to Hamming weight seven). We also prove that there are infinitely many primes $p$ with a multiplicative subgroup $A=\langle g\rangle \subset \mathbb{F}_{p}^{\ast }$, for some $g\in \{2,3,5\}$, of size $|A|\gg p/(\log p)^{3}$, where the sum–product set $A\cdot A+A\cdot A$ does not cover $\mathbb{F}_{p}$ completely.