Distinct Distances and Arithmetic Progressions
نویسنده
چکیده
According to a theorem of Szemerédi (1975), for every positive integer k and every δ > 0, there exists S = S(k, δ) such that every subset X of {1, 2, . . . , S} of size at least δS contains an arithmetic progression with k terms. Here we generalize the above result to any set of real numbers. For every positive integer k and every c > 0, there exists N = N(k, c) such that every set of n ≥ N points on the line that determine at most cn pairwise distances contains k points that form an arithmetic progression. A quantitative expression for N(k, c) is derived.
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