Abstract
Let a1, . . ., an \in \BbbR satisfy \sumi a2i = 1, and let \varepsilon 1, . . ., \varepsilon n be uniformly random \pm 1 signs and X = \sum ni=1 ai\varepsilon i. It is conjectured that X = \sum ni=1 ai\varepsilon i has Pr[X \geq 1] \geq 7/64. The best lower bound so far is 1/20, due to Oleszkiewicz. In this paper we improve this to Pr[X \geq 1] \geq 6/64.
| Original language | English |
|---|---|
| Pages (from-to) | 2393-2410 |
| Number of pages | 18 |
| Journal | SIAM Journal on Discrete Mathematics |
| Volume | 36 |
| Issue number | 3 |
| DOIs | |
| State | Published - 2022 |
Bibliographical note
Publisher Copyright:© 2022 Society for Industrial and Applied Mathematics.
Funding
\ast Received by the editors June 21, 2021; accepted for publication (in revised form) July 19, 2022; published electronically September 27, 2022. https://doi.org/10.1137/21M1428212 Funding: The first author was supported by EPSRC grant 2260624. The second author was partially supported by the Israel Science Foundation grant 1612/17. \dagger Department of Pure Maths and Mathematical Statistics, University of Cambridge, Cambridge CB2 1TN, UK ([email protected]). \ddagger Department of Mathematics, Bar Ilan University, Ramat Gan, Israel ([email protected]).
| Funders | Funder number |
|---|---|
| Engineering and Physical Sciences Research Council | 2260624 |
| Israel Science Foundation | 1612/17 |
Keywords
- Rademacher sums
- anti-concentration
- combinatorial probability