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On Infinite Sum-free Sets of Natural Numbers

✍ Scribed by Tomasz łuczak; Tomasz Schoen

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
346 KB
Volume
66
Category
Article
ISSN
0022-314X

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✦ Synopsis

A subset of the natural numbers is k-sum-free if it contains no solutions of the equation x 1 + } } } +x k = y, and strongly k-sum-free when it is l-sum-free for every l=2, ..., k. It is shown that every k-sum-free set with upper density larger than 1Â(k+1) is a subset of a periodic k-sum-free set and that each k-sum-free set with upper density larger than 2Â(k+3) is subset of a k-sum-free arithmetic progression. In particular, no k-sum-free set has upper density larger than 1Â\ 1 (k), where \ 1 (k)=min[i: i |% k&1], as conjectured by Calkin and Erdo s. Similar problems are studied also for strongly k-sum-free sets. 1997 Academic Press \ 1 (k)=min[i: i |% k&1].

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