The larger sieve and polynomial congruences

TL;DR

This paper improves Gallagher's larger sieve, extends it to higher dimensions, and provides new upper bounds for solutions to polynomial congruences, advancing understanding in sieve methods and polynomial equations.

Contribution

The authors enhance the larger sieve technique, generalize it to multiple dimensions, and derive novel bounds for polynomial congruence solutions.

Findings

Improved version of Gallagher's larger sieve.

Extension of the sieve to higher dimensions.

New upper bounds for polynomial congruence solutions.

Abstract

We obtain a small improvement of Gallagher's larger sieve and we extend it to higher dimensions. We also obtain two interesting upper bounds for the number of solutions to polynomial congruences.