On Divisors of Sums of Integers, Iii

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 a\ 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 a\ +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 a\ + h cik with a\ G A\,..., ak € Ak that is divisible by a "small" 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) < k. In order to exclude such trivial cases, we shall ask that the "small" prime factor of d\ H h ak also exceeds some prescribed bound V. In §3 we will study the case when the geometric mean of the cardinalities of the sets Axc {1,..., N} is between \fN and N. The crucial tool will be the large sieve. In §4 we will extend the range (when k > 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)