An explicit version of Bombieri's log-free density estimate and S\'ark\'ozy's theorem for shifted primes
| Autor: | Thorner, Jesse, Zaman, Asif |
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| Rok vydání: | 2022 |
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| Druh dokumentu: | Working Paper |
| Popis: | We make explicit Bombieri's refinement of Gallagher's log-free "large sieve density estimate near $\sigma = 1$" for Dirichlet $L$-functions. We use this estimate and recent work of Green to prove that if $N\geq 2$ is an integer, $A\subseteq\{1,\ldots,N\}$, and for all primes $p$ no two elements in $A$ differ by $p-1$, then $|A|\ll N^{1-1/10^{18}}$. This strengthens a theorem of S\'ark\"ozy. Comment: 24 pages. v4: Incorporates referee comments |
| Databáze: | arXiv |
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