A multiplicative analogue of Schnirelmann's theorem


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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.

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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
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All rights reserved