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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 699.10068

**Autor: ** Erdös, Paul; Sárközy, A.

**Title: ** On a conjecture of Roth and some related problems. II. (In English)

**Source: ** Number theory, Proc. 1st Conf. Can. Number Theory Assoc., Banff/Alberta (Can.) 1988, 125-138 (1990).

**Review: ** [For the entire collection see Zbl 689.00005.]

Let *A*_{1},...,*A*_{k} be a partition of the set of natural numbers into disjoint classes, and let *B* be the set of integers that have a representation in the form aa' where a,a' in *A*_{i} for some i. It is proved that for a fixed number M the minimal number of elements (over all partitions) of *B* up to M is between M(log M)^{-\alpha} and M(log M)^{-\beta} with some constants 0 < \beta < \alpha, but **sum**_{b in B}1/b > (c/k) log M with an absolute constant c > 0. The second result implies that for a fixed partition the upper density of *B* is > c/k, and it is proved that it can be < c'/k with another constant c'. The lower density is also positive, but it is proved that already for k = 3 it cannot be estimated from below from a positive constant independent of the concrete partition; for k = 2 this problem remains undecided.

The corresponding additive problem was investigated by the authors and *V. T. Sos* in Part I of this paper [Irregularities of partitions, Pap. Meet., Fertod/Hung. 1986, Algorithms. Comb. 8, 47-59 (1989; Zbl 689.10061)].

**Reviewer: ** I.Z.Ruzsa

**Classif.: ** * 11B83 Special sequences of integers and polynomials

**Keywords: ** products of integers; multiplicative representation; upper density; lower density

**Citations: ** Zbl 689.00005; Zbl 689.10061

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