Year: 2026
Type: preprint
Abstract: When $k\ge 2$ and $s>0$, we establish that for each $\varepsilon>0$, one has \[ \int_{[0,1)^{k-1}}\biggl| \sum_{1\le x\le X}e(α_2x^2+\ldots +α_k x^k)\biggr|^{2s}\,{\rm d}\boldsymbol α\ll X^{s+\varepsilon}+X^{2s-(k^2+k-2)/2}, \] confirming the main conjecture for a new family of exponential sums beyond those of Vinogradov (translation-dilation invariant) type. Our methods are based on very recent work of the author employing the nested efficient congruencing method. Consequently, analogues of our results hold also in the setting of number fields and function fields.
Source: arXiv (Cornell University)
Author: Wooley, Trevor D.
Institution: Purdue University West Lafayette
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220649783