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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
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.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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