An approximately translation-dilation invariant system

TL;DR

This paper develops a mean value estimate for exponential sums involving generalized polynomials with non-integer leading terms, extending understanding of translation-dilation invariant systems in harmonic analysis.

Contribution

It introduces a new mean value estimate for exponential sums with generalized polynomial phases, advancing the analysis of systems invariant under translation and dilation.

Findings

Established a mean value estimate for exponential sums with non-integer polynomial leading terms

Extended classical bounds to systems involving generalized polynomials with fractional degrees

Provided tools for further analysis of translation-dilation invariant systems

Abstract

Let $\theta >2$ be real and non-integral with integer part $n = \lfloor \theta \rfloor$ and let $ \phi (x)$ be a generalised polynomial with leading term $x^\theta.$ We establish a mean value estimate for the exponential sum \begin{equation*} \sum_{1 \leq x \leq P} e \left(\alpha_1 x + \cdots + \alpha_n x^n + \alpha_\phi \phi (x) \right). \end{equation*}