The Erdős discrepancy problem

Abstract

We show that for any sequence f (1), f (2), . . . taking values in {−1, +1}, the discrepancyof f is infinite. This answers a question of Erdős. In fact the argument also applies to sequences f taking values in the unit sphere of a real or complex Hilbert space.The argument uses three ingredients. The first is a Fourier-analytic reduction, obtained as part of the Polymath5 project on this problem, which reduces the problem to the case when f is replaced by a (stochastic) completely multiplicative function g. The second is a logarithmically averaged version of the Elliott conjecture, established recently by the author, which effectively reduces to the case when g usually pretends to be a modulated Dirichlet character. The final ingredient is (an extension of) a further argument obtained by the Polymath5 project which shows unbounded discrepancy in this case.

The Logarithmically Averaged Chowla and Elliott Conjectures for Two-Point Correlations

“…Readers who are interested just in the case of the Liouville function (Theorem 2) can skip the initial reductions and move directly (for the application to the Erdős discrepancy problem in [30], one only needs the special case when g 2 = g 1 and g 1 is completely multiplicative and takes values in S 1 . In that case one can also move directly to Theorem 11, skipping the initial reductions) to Theorem 11 below.…”

“…In a subsequent paper [30], we will combine Theorem 3 with some arguments arising from the Polymath5 project [27] to obtain an affirmative answer to the Erdős discrepancy problem [5]: If E is a statement, we use 1 E to denote the indicator, thus 1 E = 1 when E is true and 1 E = 0 when E is false.…”

A note on multiplicative functions resembling the Möbius function

“…The solution of the Erdős discrepancy problem (see [9]) implies that a completely multiplicative function f : N → {−1, 1} has unbounded partial sums. However, a completely multiplicative function f : N → {−1, 0, 1} may have bounded partial sums, for instance, a real non-principal Dirichlet character χ.…”

Correlations of multiplicative functions along deterministic and independent sequences

“…Henceforth, with S we denote the complex unit circle. Using the previous result for ℓ = 2 and f 1 = f , f 2 =f , where f : N → S is a strongly aperiodic multiplicative function, we can immediately deduce using an argument from [24] (see [6,Proposition 2.3] for the needed result) the following: Remark. The same argument works if f coincides with a strongly aperiodic multiplicative function f : N → U on a set with logarithmic density one and satisfies lim inf N →∞ E log n∈[N ] |f (n)| > 0.…”