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**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 N^{2} 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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