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Tight bound for the density of sequence of integers the sum of no two of which is a perfect square. (English) Zbl 1066.11004
Summary: P. Erdős and A. Sárközy [Bull. Greek Math. Soc. 18, 204–223 (1977; Zbl 0413.10049), p. 209] proposed the problem of determining the maximal density attainable by a set \(S\) of positive integers having the property that no two distinct elements of \(S\) sum up to a perfect square. J.-P. Massias [Sur les suites dont les sommes des terms 2 à 2 ne sont pars des carrés. Publications du Département de mathématiques de Limoges (1982)] exhibited such a set consisting of all \(x\equiv 1 \pmod 4\) with \(x\equiv14,26,30 \pmod {32}\). J. C. Lagarias et al. [J. Combin. Theory, Ser. A 33, 167–185 (1982; Zbl 0489.10052)] showed that for any positive integer \(n\), one cannot find more than \(11n/25\) residue classes (mod \(n\)) such that the sum of any two is never congruent to a square (mod \(n\)), thus essentially proving that the Massias’ set has the best possible density. They [J. Combin. Theory, Ser. A 34, 123–129 (1983; Zbl 0514.10041)] also proved that the density of such a set S is never \(>0.475\) when we allow general sequences. We improve on the lower bound for general sequences, essentially proving that it is not \(0.475\), but arbitrarily close to Image, the same as that for sequences made up of only arithmetic progressions.

MSC:
11B05 Density, gaps, topology
11B75 Other combinatorial number theory
11L03 Trigonometric and exponential sums, general
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