$\ell^p(\mathbb{Z}^n)$-estimate for long $r$-variational seminorm of discrete Birch-Magyar averages

TL;DR

This paper establishes $ ext{ell}^p( ext{Z}^n)$-bounds for long $r$-variational seminorms of discrete averages, advancing understanding of their behavior and applications in ergodic theory.

Contribution

It provides new $ ext{ell}^p$ estimates for long $r$-variational seminorms of discrete Birch-Magyar and algebraic variety averages, extending previous results.

Findings

Proved $ ext{ell}^p$ estimates for discrete Birch-Magyar averages.

Established bounds for averages over algebraic varieties.

Applied results to ergodic theory contexts.

Abstract

We prove $\ell^p(\mathbb{Z}^n)-$estimates for long $r$-variational seminorm of two families of averages: discrete Birch-Magyar averages, for $r>max\{p,p'\}$ with $p>\frac{2c_{\mathfrak{R}}-2}{2c_{\mathfrak{R}}-3}$ and discrete Hardy-Littlewood type averages over certain algebraic varieties, for $r>max\{p,p'\}$ with $p>1$. Further, we discuss an application of these results in ergodic theory.