On simultaneous approximations to the logarithms of primes

M. A. Korolev ; I. S. Rezvyakova

Čebyševskij sbornik, Tome 23 (2022) no. 5, pp. 87-100

Résumé

In the first part of the paper, a modification of elementary Titchmarsh's method is applied to the proof of the local Kronecker's theorem. For any finite real sequence $\boldsymbol{\bar{\lambda}} = (\lambda_{1},\ldots,\lambda_{r})$ of linearly independent (over $\mathbb{Q}$) numbers and for any $\varepsilon>0$, this method leads to the explicit upper bound of the value $h = h(\varepsilon,\boldsymbol{\bar{\lambda}})$ with the following property: for any real sequence $\boldsymbol{\bar{\alpha}} = (\alpha_{1},\ldots,\alpha_{r})$, any interval of the length $h$ contains a point $t$ such that $\|t\lambda_{s}-\alpha_{s}\|\leqslant\varepsilon$, $1\leqslant s\leqslant r$. Such estimate is weaker than the best known, but it's proof is quite simple and leads to the same (in essence) results in the applications. The second part contains the short memoirs concerning the academician Alexey Nikolaevich Parshin who passed away on June, 18 this year.

Détail

Keywords: local Kronecker's theorem, simultaneous approximations, logarithms of primes, squarefree numbers.