### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 224-235 |

Number of pages | 12 |

Journal | Bulletin of the Australian Mathematical Society |

Volume | 94 |

Issue number | 2 |

DOIs | |

Publication status | Published - 23 May 2016 |

### Fingerprint

### Keywords

- additive bases
- distribution of integers with multiplicative constraints
- primes
- sumsets
- Waring’s problem and variants

### ASJC Scopus subject areas

- Mathematics(all)

### Fields of Expertise

- Information, Communication & Computing

### Cite this

**ALMOST ALL PRIMES HAVE A MULTIPLE OF SMALL HAMMING WEIGHT.** / ELSHOLTZ, CHRISTIAN.

Research output: Contribution to journal › Article › Research › peer-review

*Bulletin of the Australian Mathematical Society*, vol. 94, no. 2, pp. 224-235. https://doi.org/10.1017/S000497271600023X

}

TY - JOUR

T1 - ALMOST ALL PRIMES HAVE A MULTIPLE OF SMALL HAMMING WEIGHT

AU - ELSHOLTZ, CHRISTIAN

PY - 2016/5/23

Y1 - 2016/5/23

N2 - 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.

AB - 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.

KW - additive bases

KW - distribution of integers with multiplicative constraints

KW - primes

KW - sumsets

KW - Waring’s problem and variants

UR - http://www.scopus.com/inward/record.url?scp=84969804473&partnerID=8YFLogxK

U2 - 10.1017/S000497271600023X

DO - 10.1017/S000497271600023X

M3 - Article

VL - 94

SP - 224

EP - 235

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

SN - 0004-9727

IS - 2

ER -