论文标题
二维WEYL和沿线取消方形取消的Weyl总和
Two-dimensional Weyl sums failing square-root cancellation along lines
论文作者
论文摘要
我们表明，在几乎所有单位圆环的所有线性切片上，一定的二维Weyl weyl sum of Lengum $ p $的weyl族都具有$ p^{3/4 + o（1）} $的值，这与人们普遍认为Weyl Sugs应在单位torus的通用子类别上表现出平方root root的期望相矛盾。这是J. Brandes，S。T。Parsell，C。Poulias，G。Shakan和R. C. Vaughan（2020年）的结果，从二次和立方单一元素到任意程度的一般多项式。我们方法的新成分是E. Bombieri（1966）沿曲线的指数总和的经典结果，以及R. J. Duffin和A. C. Schaeffer（1941）通过具有主要分母的合理数字对Diophantine近似值。
We show that a certain two-dimensional family of Weyl sums of length $P$ takes values as large as $P^{3/4 + o(1)}$ on almost all linear slices of the unit torus, contradicting a widely held expectation that Weyl sums should exhibit square-root cancellation on generic subvarieties of the unit torus. This is an extension of a result of J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan (2020) from quadratic and cubic monomials to general polynomials of arbitrary degree. The new ingredients of our approach are the classical results of E. Bombieri (1966) on exponential sums along a curve and R. J. Duffin and A. C. Schaeffer (1941) on Diophantine approximations by rational numbers with prime denominators.
