Almost all primes have a multiple of small Hamming weight

TL;DR

This paper improves bounds on the Hamming weight of multiples of primes, showing that a multiple of almost all primes can be expressed with only six ones in binary, and explores properties of certain prime-related multiplicative subgroups.

Contribution

It reduces the known Hamming weight bound for multiples of primes from 16 or 66 to 6, and demonstrates the existence of infinitely many primes with specific subgroup and sum-product properties.

Findings

Almost all primes have a multiple with Hamming weight 7.

Existence of infinitely many primes with large subgroups where sum-product sets do not cover the entire field.

Improved bounds on binary representations of prime multiples.

Abstract

Recent results of Bourgain and Shparlinski imply that for almost all primes $p$ there is a multiple $mp$ that can be written in binary as $mp= 1+2^{m_1}+ \cdots +2^{m_k}, \quad 1\leq m_1 < \cdots < m_k,$ with $k=66$ or $k=16$, respectively. We show that $k=6$ (corresponding to Hamming weight $7$) suffices. We also prove there are infinitely many primes $p$ with a multiplicative subgroup $A=<g>\subset \mathbb{F}_p^*$, 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.