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A multiplicative analogue of Schnirelmann's theorem

Accepted version
Peer-reviewed

Type

Article

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Abstract

The classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper, we consider the analogous multiplicative setting of the cyclic group (ℤ/𝑞ℤ)× and prove a similar result. For all suitably large primes 𝑞 we define 𝑃𝜂 to be the set of primes less than 𝜂𝑞 , viewed naturally as a subset of (ℤ/𝑞ℤ)× . Considering the 𝑘 ‐fold product set 𝑃(𝑘)𝜂={𝑝1𝑝2⋯𝑝𝑘:𝑝𝑖∈𝑃𝜂} , we show that, for 𝜂≫𝑞−1/4+𝜖, there exists a constant 𝑘 depending only on 𝜖 such that 𝑃(𝑘)𝜂=(ℤ/𝑞ℤ)× . Erdös conjectured that, for 𝜂=1, the value 𝑘=2 should suffice: although we have not been able to prove this conjecture, we do establish that 𝑃(2)1 has density at least 164(1+𝑜(1)) . We also formulate a similar theorem in almost‐primes, improving on existing results.

Description

Keywords

math.NT, math.NT, math.CO

Journal Title

Bulletin of the London Mathematical Society

Conference Name

Journal ISSN

0024-6093
1469-2120

Volume Title

48

Publisher

Wiley

Rights

All rights reserved