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On sums and products of integers. (English) Zbl 0526.10011

Studies in Pure Mathematics, Mem. of P. Turán, 213-218 (1983).
[This article was published in the book announced in Zbl 0512.00007.]
Denoting by \(f(n)\) the largest integer such that for every \(\{1\leq a_1\leq\dots\leq a_n\}\) integer set there are at least \(f(n)\) distinct numbers of the form \(a_i+a_j\), \(a_ia_j\), \(1\leq i\leq j\leq n\), the authors prove that \[ n^{1+c_1}< f(n)< n^2\exp(-c_2\log n/\log\log n). \] Some other related results and a lot of related conjectures are also discussed. The proof is self-contained and based only on elementary combinatorial arguments.

MSC:

11B75 Other combinatorial number theory
11B30 Arithmetic combinatorics; higher degree uniformity

Citations:

Zbl 0512.00007