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The Erdos discrepancy problem
| Content Provider | Semantic Scholar |
|---|---|
| Author | Tao, Terence |
| Copyright Year | 2015 |
| 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 Erd\H{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. |
| Starting Page | 609 |
| Ending Page | 609 |
| Page Count | 1 |
| File Format | PDF HTM / HTML |
| DOI | 10.19086/da.609 |
| Alternate Webpage(s) | https://www.mathnet.or.kr/real/2017/06/discrepancy_problem.pdf |
| Alternate Webpage(s) | http://export.arxiv.org/pdf/1509.05363 |
| Alternate Webpage(s) | https://arxiv.org/pdf/1509.05363v6.pdf |
| Alternate Webpage(s) | https://doi.org/10.19086/da.609 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
