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

TL;DR

The paper investigates long $r$-variational bounds for two families of discrete averages on ${\mathbb Z}^n$: Birch-Magyar averages and Hardy-Littlewood type averages over algebraic varieties. It establishes $\ell^p$-boundedness of $V_r$ for lacunary parameter sequences when $r>\max\{p,p'\}$ and $p$ lies within Birch-rank dependent ranges, using a discrete circle-method decomposition and jump inequalities to reduce to $p=2$, followed by square-function and Littlewood-Paley-type analyses. For the algebraic-variety averages, analogous $\ell^p$-variation bounds are obtained under regularity of the form and lacunarity, aided by Weyl-sum estimates and decay properties of surface-measure Fourier transforms. The results further extend to ergodic theory via a transference principle, yielding variational bounds for ergodic Birch-Magyar averages with the same $r$-range, and thereby providing a robust link between discrete harmonic analysis and ergodic theory. The work advances understanding of distributional and variational behavior of discrete averages on surfaces and their ergodic counterparts, with sharp $p=2$-type sharpness in the variation range.

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.

Theorems & Definitions (13)

• Theorem 1.1
• Lemma 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Example 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Proposition 2.1