Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  780.11040
Autor:  Erdös, Paul; Pomerance, C.; Sárközy, A.; Stewart, C.L.
Title:  On elements of sumsets with many prime factors. (In English)
Source:  J. Number Theory 44, No.1, 93-104 (1993).
Review:  Let \nu(n) be the number of distinct prime factors of n. The following problem is studied in the paper. Having two finite sets of positive integers A and B how big is \nu(n) on the sumset A+B? Suppose that A and B are subsets of {n \leq N/2}. Then certainly max\nu(a+b) \leq m where m = m(N) is the maximal value of \nu(n) for n \leq N. It is shown that for dense sets this upper bound is almost attained, more precisely, for each \epsilon > 0 there is a c(\epsilon) such that if |A| |B| > \epsilon N2 then we have max\nu(a+b) > m- c(\epsilon) \sqrt{m}. It is also shown that this result is close to best possible. The proof has both probabilistic and combinatorial flavour.
Reviewer:  A.Balog (Budapest)
Classif.:  * 11N25 Distribution of integers with specified multiplicative constraints
                   11B75 Combinatorial number theory
                   11N56 Rate of growth of arithmetic functions
Keywords:  hybrid theorems; multiplicative properties of sumsets

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