On the Erdős Discrepancy Problem

According to the Erdős discrepancy conjecture, for any infinite ±1 sequence, there exists a homogeneous arithmetic progression of unbounded discrepancy. In other words, for any ±1 sequence (x1, x2, ...) and a discrepancy C, there exist integers m and d such that | ∑ m i=1 xi·d| > C. This is an 80-year-old open problem and recent development proved that this conjecture is true for discrepancies up to 2. Paul Erdős also conjectured that this property of unbounded discrepancy even holds for the restricted case of completely multiplicative sequences (CMSs), namely sequences (x1, x2, ...) where xa·b = xa · xb for any a, b ≥ 1. The longest CMS with discrepancy 2 has been proven to be of size 246. In this paper, we prove that any completely multiplicative sequence of size 127, 646 or more has discrepancy at least 4, proving the Erdős discrepancy conjecture for CMSs of discrepancies up to 3. In addition, we prove that this bound is tight and increases the size of the longest known sequence of discrepancy 3 from 17, 000 to 127, 645. Finally, we provide inductive construction rules as well as streamlining methods to improve the lower bounds for sequences of higher discrepancies. Introduction Discrepancy theory addresses the problem of distributing points uniformly over some geometric object, and studies how irregularities inevitably occur in these distributions. For example, this subfield of combinatorics aims to answer the following question: for a given set U of n elements, and a finite family S = {S1, S2, . . . , Sm} of subsets of U , is it possible to color the elements of U in red or blue, such that the difference between the number of blue elements and red elements in any subset Si is small? Important contributions in discrepancy theory include the Beck-Fiala theorem [1] and Spencer’s Theorem [2]. The Beck-Fiala theorem guarantees that if each element appears at most t times in the sets of S, the elements can be colored so that the imbalance, or discrepancy, is no more than 2t − 1. According to the Spencer’s theorem, the discrepancy of S grows at most as Ω( √ n log(2m/n)). Nevertheless, some important questions remain open. According to Paul Erdős himself, two of his oldest conjectures relate to the discrepancy of homogeneous arithmetic progressions (HAPs) [3]. Namely, a HAP of length k and of common difference d corresponds to the sequence (d, 2d, . . . , kd). The first conjecture can be formulated as follows: *Submitted on April 14, 2014 to the 20th International Conference on Principles and Practice of Constraint Programming.