# Bounds for Lacunary maximal functions given by Birch--Magyar averages

## TL;DR

This paper investigates lacunary discrete maximal functions associated with hypersurfaces from Diophantine equations, providing new bounds and contrasting positive and negative results in the context of harmonic analysis.

## Contribution

It introduces improved bounds for lacunary maximal functions and develops new techniques for analyzing main terms near , advancing understanding of these operators.

## Key findings

- Positive bounds for lacunary maximal functions are improved.
- Negative results highlight differences from nonsingular hypersurface cases.
- A new bound on main terms near  is established.

## Abstract

We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this problem differs substantially from that of lacunary discrete maximal operators defined along a nonsingular hypersurface. Our positive results are improvements over bounds for the corresponding full maximal functions which were initially studied by Magyar. In order to obtain positive results, we use an interpolation technique of the second author to reduce problem to a maximal function of main terms. The main terms take the shape of those introduced in work of the first author, which is a more localized version of the main terms that appear in work of Magyar. The main ingredient of this paper is a new bound on the main terms near $\ell^1$. For our negative results we generalize an argument of Zienkiewicz.