Abstract
It is proved that given ε>0, there is δ(ε)>0 such that ifS is a measurable set of [0,N], |S|>εN, then there is a triplex, x+h, x+h 2 inS withh satisfyingh>δ(ε)N 1/2. The argument is related to [B] and uses the behavior of certain non-linear convolution-type operators. The method can be adapted in a variety of situations. For instance, it can be used to prove the analogue of the previous statement with the square replaced by another power,h 3,h 4 etc.
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References
[B] J. Bourgain,A Szemerédi type theorem for sets of positive density in R k, Isr. J. Math.54 (1986), 307–316.
[B2] J. Bourgain,On square functions on the trigonometric system, Bull. Soc. Math Belg., Ser. B (1985), 20–26.
[F] H. Furstenberg, Seminar talk at Stanford University, August 1986.
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Bourgain, J. A nonlinear version of Roth's theorem for sets of positive density in the real line. J. Anal. Math. 50, 169–181 (1988). https://doi.org/10.1007/BF02796120
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DOI: https://doi.org/10.1007/BF02796120