Furstenberg, Harry Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions. (English) Zbl 0347.28016 J. Anal. Math. 31, 204-256 (1977). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 53 ReviewsCited in 332 Documents MathOverflow Questions: What are some examples of proving that a thing exists by proving that the set of such things has positive measure? MSC: 28D05 Measure-preserving transformations 47A35 Ergodic theory of linear operators 54H20 Topological dynamics (MSC2010) 11B83 Special sequences and polynomials PDFBibTeX XMLCite \textit{H. Furstenberg}, J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016) Full Text: DOI Online Encyclopedia of Integer Sequences: Size of the largest subset of {1,2,...,n} such that no two distinct elements differ by a perfect square. Consider all sequences of n distinct positive integers for which no two different elements have a difference which is a square. This sequence gives the smallest maximal integer in such sequences. References: [1] Furstenberg, H., The structure of distal flows, Amer. J. Math., 85, 477-515 (1963) · Zbl 0199.27202 [2] Khintchine, A. Y., Three Pearls of Number Theory (1952), New York: Graylock Press, New York · Zbl 0048.27202 [3] Krengel, U., Weakly wandering vectors and weakly independent partitions, Trans. Amer. Math. Soc., 164, 199-226 (1972) · Zbl 0205.13903 [4] Mackey, G. W., Induced representation of locally compact groups, I, Ann. of Math., 55, 101-139 (1952) · Zbl 0046.11601 [5] Roth, K. F., Sur quelques ensembles d’entiers, C. R. Acad. Sci. Paris, 234, 388-390 (1952) · Zbl 0046.04302 [6] Szemerédi, E., On acts of integers containing no four elements in arithmetic progression, Acta Math. Acad. Sci. Hungar., 20, 89-104 (1969) · Zbl 0175.04301 [7] Szemerédi, E., On acts of integers containing no k elements in arithmetic progression, Acta Arith., 27, 199-199 (1975) · Zbl 0303.10056 [8] R. Zimmer,Ergodic actions with generalized discrete spectrum, to appear. · Zbl 0349.28011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.