On the $j$-th smallest modulus of a covering system with distinct moduli
| Autor: | Klein, Jonah, Koukoulopoulos, Dimitris, Lemieux, Simon |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Covering systems were introduced by Erd\H{o}s in 1950. In the same article where he introduced them, he asked if the minimum modulus of a covering system with distinct moduli is bounded. In 2015, Hough answered affirmatively this long standing question. In 2022, Balister, Bollob\'as, Morris, Sahasrabudhe and Tiba gave a simpler and more versatile proof of Hough's result. Building upon their work, we show that there exists some absolute constant $c>0$ such that the $j$-th smallest modulus of a minimal covering system with distinct moduli is $\le \exp(cj^2/\log(j+1))$. Comment: 8 pages, minor corrections and changes. Final version, to appear in Int. J. Number Theory |
| Databáze: | arXiv |
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