On divisors of sums of integers. III

Abstract

1. Let P{ή) and p(n) denote the greatest and smallest prime factor of n, respectively. Recently in several papers, Balog, Erdόs, Maier, Sarkozy, and Stewart have studied problems of the following type: if A\,...,Ak are dense sets of positive integers, then what can be said about the arithmetical properties of the sums H h % with a, G A\,...,ak G Akl In particular, Balog and Sarkόzy proved that there is a sum a\+ci2 (#i G A\9 aι G A2) for which P(a\+aι) is small, i.e., all the prime factors of +aι are small. On the other hand, Balog and Sarkόzy and Sarkόzy and Stewart studied the existence of a sum CL Λ h % for which P(a H h ak) is large. In this paper we study p(a H \ak). Our goal is to show that if A\,..., Ak are sets of positive integers then there exists a sum + h cik with G A\,..., ak € Ak that is divisible by a prime. In the most interesting special case, namely A = = Ak, there are sums CL Λ Vak divisible by /c, so that p(a H h ak) 2) by studying the case min, \At > N /+. Here Gallagher's larger sieve will be used. The results in §§3 and 4 do not give especially good results when the sets A\,...,Ak are very dense. In §5, we will give an essentially best possible result for the small prime factors of the sums d Λ h ak in the case when (\AX \Ak\) / > Nexp(-clogklogN/loglogN)