The Erdős discrepancy problem

Terence Tao

doi:10.19086/da.609

Abstract

The Erdős discrepancy problem, Discrete Analysis 2016:1, 27 pp. One of Erdős's most famous problems was his _discrepancy_ problem, which is the following deceptively simple question. Let $\epsilon_1,\epsilon_2,\dots$ be a sequence of 1s and -1s and let $m$ be a positive integer. Must there exist positive integers $n$ and $d$ such that $|\sum_{i=1}^n\epsilon_{id}|\geq m$? This would tell us that when we restrict to the arithmetic progression $\{d,2d,\dots,nd\}$ the number of 1s and the number of -1s differ by at least $m$ -- hence the word "discrepancy". A simple observation is that if the sequence $(\epsilon_i)$ is completely multiplicative (that is, $\epsilon_{ij}=\epsilon_i\epsilon_j$ for every $i$ and $j$), then the question reduces to asking whether the partial sums $\sum_{i=1}^n\epsilon_i$ must be unbounded. Even this special case is surprisingly hard. Very roughly, Tao's strategy can be described as follows. First, he reduces the general problem to one that is closely related to the special case just described. Next, he proves, using a recent result of his that he calls a logarithmically averaged nonasymptotic Elliott conjecture, that any multiplicative counterexample to the Erdõs discrepancy problem would have to be of a very special form that would make it similar to a Dirichlet character. And finally, he proves that multiplicative functions that resemble Dirichlet characters must give rise to at least logarithmic growth on average in the partial sums. The second step is related to a recent breakthrough of Matomäki and Radziwiłł, who established results about correlations of successive values of certain multiplicative functions. Note that if one can show that there is very little correlation between nearby values of such functions, then one has shown that they are not counterexamples to the Erdõs discrepancy problem, since one cannot keep the partial sums bounded without significant local correlation. Before the work of Matomäki and Radziwiłł it was thought that saying anything about such correlations was completely out of reach, but with their work and subsequent extensions by Tao the picture has changed dramatically, to the point where it is possible to contemplate an argument such as the one in this paper. <div class="flex-video"> <iframe width="560" height="315" src="https://www.youtube.com/embed/QauoO0j9Y9Y" frameborder="0" allowfullscreen></iframe> </div>

Highlights

The discrepancy is the largest magnitude of a sum of f along homogeneous arithmetic progressions {d, 2d, . . . , nd} in the natural numbers N = {1, 2, 3, . . . }
Every sequence f (1), f (2), . . . taking values in {−1, +1} has infinite discrepancy. This answers a question of Erdos [9], which was recently the subject of the Polymath5 project [19]; see the recent report [10] on the latter project for further discussion
A similar sequence with N = 130 000 of discrepancy 3 was constructed in that paper, as well as a sequence with N = 127 645 of discrepancy 3 that was the restriction to {1, . . . , N} of a completely multiplicative sequence taking values in {−1, +1}. This slow growth in discrepancy may be compared with the log N type divergence in Example 1.5

Summary

Introduction

Given a sequence f : N → H taking values in a real or complex Hilbert space H, define the discrepancy of f to be the quantity n sup ∑ f ( jd). N} of a completely multiplicative sequence taking values in {−1, +1} (with the latter value of 127 645 being the best possible value of N; see [1] for a separate √computation confirming this threshold) This slow growth in discrepancy may be compared with the log N type divergence in Example 1.5. It remains to demonstrate Theorem 1.8 for random completely multiplicative functions g that obey (1.4) with high probability for large X and small ε Such functions g can be viewed as (somewhat complicated) generalisations of the Borwein-Choi-Coons example (Example 1.4), and it turns out that a more complicated version of the analysis in Example 1.4 (or Example 1.5) suffices to establish a lower bound for E| ∑nj=1 g( j)|2 (of logarithmic type, similar to that in Example 1.5) which is enough to conclude Theorem 1.8 and Theorem 1.1 and Corollary 1.2. We do not know if there is any deeper significance to this similarity

Notation

Fourier analytic reduction

Applying the Elliott-type conjecture

A generalised Borwein-Choi-Coons analysis

Sums of multiplicative functions