Sharp mean value estimates in the absence of translation-dilation invariance

Abstract

We establish sharp mean value estimates for various exponential sums lacking translation-dilation invariance properties. For example, suppose that $k\ge 1$ and $s>0$. Then, for each $\varepsilon>0$, one has \[ \int_{[0,1)^k}\biggl| \sum_{1\le h\le X}\sum_{1\le z\le X}e(hz\alpha_1+\ldots +hz^k\alpha_k)\biggr|^{2s}\,{\rm d}\boldsymbol \alpha \ll X^\varepsilon\bigl( X^s+X^{2s-k(k+3)/2}\bigr), \] confirming the main conjecture for a new family of exponential sums. Our methods are based on the application of very recent work of the author employing the nested efficient congruencing method.