Discrepancy in arithmetic progressions
نویسندگان
چکیده
منابع مشابه
Discrepancy in Arithmetic Progressions
It is proven that there is a two-coloring of the first n integers forwhich all arithmetic progressions have discrepancy less than const.n1/4. Thisshows that a 1964 result of K. F. Roth is, up to constants, best possible. Department of Applied Mathematics, Charles University, Malostranské nám. 25,118 00 Praha 1, Czech RepublicE-mail address: [email protected] Courant Insti...
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Estimating the discrepancy of the set of all arithmetic progressions in the first N natural numbers was one of the famous open problem in combinatorial discrepancy theory for a long time, successfully solved by K. Roth (lower bound) and Beck (upper bound). They proved that D(N) = minχ maxA | ∑ x∈A χ(x)| = Θ(N1/4), where the minimum is taken over all colorings χ : [N ] → {−1, 1} and the maximum ...
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The set system of all arithmetic progressions on [n] is known to have a discrepancy of order n1/4. We investigate the discrepancy for the set system S3 n formed by all sums of three arithmetic progressions on [n] and show that the discrepancy of S3 n is bounded below by Ω(n1/2). Thus S3 n is one of the few explicit examples of systems with polynomially many sets and a discrepancy this high.
متن کاملDiscrepancy of Sums of two Arithmetic Progressions
Estimating the discrepancy of the hypergraph of all arithmetic progressions in the set [N ] = {1, 2, . . . , N} was one of the famous open problems in combinatorial discrepancy theory for a long time. An extension of this classical hypergraph is the hypergraph of sums of k (k ≥ 1 fixed) arithmetic progressions. The hyperedges of this hypergraph are of the form A1 + A2 + . . . + Ak in [N ], wher...
متن کاملDiscrepancy of Cartesian Products of Arithmetic Progressions
We determine the combinatorial discrepancy of the hypergraph H of cartesian products of d arithmetic progressions in the [N ]d–lattice ([N ] = {0, 1, . . . ,N − 1}). The study of such higher dimensional arithmetic progressions is motivated by a multi-dimensional version of van der Waerden’s theorem, namely the Gallai-theorem (1933). We solve the discrepancy problem for d–dimensional arithmetic ...
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ژورنال
عنوان ژورنال: Journal of the American Mathematical Society
سال: 1996
ISSN: 0894-0347,1088-6834
DOI: 10.1090/s0894-0347-96-00175-0