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Arithmetic progressions in subset sums

✍ Scribed by P. Erdős; A. Sárközy

Publisher
Elsevier Science
Year
1992
Tongue
English
Weight
693 KB
Volume
102
Category
Article
ISSN
0012-365X

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✦ Synopsis

~roughout this paper we use the following notatians: The cardinality of the finite set Y is denoted by ISI -.s& B8, . I s den&e finite or infinite sets of positive integers. If & is a finite or infinite set of positive integers, then S(d) denotes the set of the distinct positive integers n that can be represented in the form n = Coed &*a where E, = 0 or 1 for all a (and if JZZ is infinite, then all but finitely many of the E'S are equal to 0). 2. Let u = F(N, t) denote the greatest integer u. such that for every d c (II 2, -. . ~ IV) with I.&l = t, the set P(d) contains 11 consecutive multiples of a positive integer d: ((x + l)d, (x + 2)d, . _ . , (x + u)d) c P(a) for some x and d, and let v = G(N, t) denote the greatest integer v such that for every tit (17 2, . f -, N} with l&j = t, the set 9(a) contains an arithmetic progression of length IJ: {y + (z -I-l)dl y + (z + 2)d, . . . , y + (z -I-u)d} c P"(d) for some y, t and d(>O). Clearly, F(N, t) 6 G(iV, t) for all N, t. S&-k&y [5] proved that (G(N, t) 2 )F(N, t) > 8-110-4tz for N > &, t B l(H)(N log N)ln.

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