ALMOST ALL PRIMES HAVE A MULTIPLE OF SMALL HAMMING WEIGHT

We improve recent results of Bourgain and Shparlinski to show that, for almost all primes (Formula presented.), there is a multiple (Formula presented.) that can be written in binary as (Formula presented.) with (Formula presented.) (corresponding to Hamming weight seven). We also prove that there are infinitely many primes (Formula presented.) with a multiplicative subgroup (Formula presented.), for some (Formula presented.), of size (Formula presented.), where the sum–product set (Formula presented.) does not cover (Formula presented.) completely.