An approximately translation-dilation invariant system

Abstract

Let $\theta >2$ be real and non-integral with integer part $n=\lfloor \theta \rfloor$ and let $\varphi (x)$ 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(α_1 x + \cdots + α_n x^n + α_ϕϕ(x) \right). \end{equation*}

The proof contains two gaps. Thank you to an anonymous referee for pointing these out. It seems that (if one were to fix these flaws) this approach would yield a weaker result than the one claimed in the manuscript. An updated version, fixing these flaws, might appear in the future