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- W1837210137 abstract "We show that for any sequence $f: {bf N} to {-1,+1}$ taking values in ${-1,+1}$, the discrepancy $$ sup_{n,d in {bf N}} left|sum_{j=1}^n f(jd)right| $$ of $f$ is infinite. This answers a question of ErdH{o}s. 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 ${bf g}$. The second is a logarithmically averaged version of the Elliott conjecture, established recently by the author, which effectively reduces to the case when ${bf g}$ 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." @default.
- W1837210137 created "2016-06-24" @default.
- W1837210137 creator A5066728260 @default.
- W1837210137 date "2015-09-17" @default.
- W1837210137 modified "2023-10-16" @default.
- W1837210137 title "The Erdos discrepancy problem" @default.
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- W1837210137 doi "https://doi.org/10.48550/arxiv.1509.05363" @default.
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