Sums of positive density subsets of the primes

• Autorzy: Kaisa Matomäki
• Języki publikacji: EN

Abstrakty

EN

We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least $(1-o(1))α/(e^{γ} log log (1/β))$, which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of $ℤ*_m$ using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any $A, B ⊆ ℤ*_m$ of densities α and β, the density of A+B in $ℤ_m$ is at least $(1-o(1))α/(e^{γ} log log (1/β))$, which is asymptotically best possible when m is a product of small primes. We also discuss an inverse question.

Kategorie tematyczne

• 11P32: Goldbach-type theorems; other additive questions involving primes
• 11B30: Arithmetic combinatorics; higher degree uniformity

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 159
• Numer: 3
• Strony: 201-225

Daty

wydano

2013

Twórcy

autor

Kaisa Matomäki

• Department of Mathematics, University of Turku, 20014 Turku, Finland

Identyfikatory

DOI

10.4064/aa159-3-1