Discrepancy of (centered) arithmetic progressions in [formula omitted

Abstract

One of the celebrated and deep theorems in discrepancy theory says that the 2-color discrepancy of the hypergraph of arithmetic progressions in the first n integers is Θ(n1/4) (lower bound by Roth (1964), upper bound Matoušek, Spencer (1996)). In this paper we consider the problem of finding the c-color discrepancy, c≥2, of arithmetic progressions and centered arithmetic progressions resp. in Zp. We will show that its discrepancy is much larger than for arithmetic progressions in the first n integers. It is easy to prove that for p=2 the discrepancy of the arithmetic progressions in Zp is exactly (c−1)/c and it is 1 for any p and c=2 for the centered arithmetic progressions. But no matching upper and lower bounds were known beyond these special cases. We show that for both hypergraphs the lower bound is tight up to a logarithmic factor in p: the lower bound is Ω(pc) and the upper bound is O(pclnp). The main work is the proof of the lower bounds with Fourier analysis on Zp. We consider the result for the centered arithmetic progressions as our main achievement because here due to the lack of translation-invariance of the hypergraph Fourier analysis alone does not work and has to be combined with a suitable decomposition of such progressions. Techniques of this type may be useful also in other areas where lower bounds for combinatorial functions are sought.