Discrepancy of Centered Arithmetic Progressions in Zp (Extended Abstract)

Abstract

Combinatorial discrepancy theory is the theory of coloring hypergraphs in a balanced way, generalizing graph coloring, is an important field in combinatorics with applications to algorithms and complexity, such as efficient sampling, randomization and derandomization. One of the most 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 ( n 1 / 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 Z p . It is notable that the problem of approximating the discrepancy of centered arithmetic progressions in the natural numbers is certainly one of the most difficult problems posed by P. Erdös, with almost no progress over the last 30 years. The problem considered in this paper can be seen as a kind of non-trivial relaxation of the Erdös problem. It is easy to prove that for p = 2 the discrepancy of the arithmetic progressions in Z p 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 ( p c ) and the upper bound is O ( p c ln p ) . The main work is the proof of the lower bounds with Fourier analysis on Z p . 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 new decomposition of such progressions. Techniques of this type may be usefull also in other areas where lower bounds for combinatorial functions are sought.