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+2m1β+β―+2mkβ,1β€m1β<β―<mkβ, 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=βFpββ, for some
gβ{2,3,5}, of size β£Aβ£β«p/(logp)3, where the sum-product set
Aβ A+Aβ A does not cover Fpβ completely