Hybrid bounds on two-parametric family Weyl sums along smooth curves

On discrete fractional integral operators and mean values of Weyl Sums

In this paper we prove new $(\ell^p, \ell^q)$ bounds for a discrete fractional integral operator by applying techniques motivated by the circle method of Hardy and Littlewood to the Fourier multiplier of the operator. From a different perspective, we describe explicit interactions between the Fourier multiplier and mean values of Weyl sums. These mean values express the average behaviour of the number $r_{s,k}(l)$ of representations of a positive integer $l$ as a sum of $s$ positive $k$-th powers. Recent deep results within the context of Waring's problem and Weyl sums enable us to prove a fur…

New bounds of Weyl sums

We augment the method of Wooley (2015) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also extend these results and ideas to principally new and very general settings of arbitrary orthogonal projections of the vectors of the coefficients $(u_1, \ldots , u_d)$ onto a lower dimensional subspace. This new point of view has an additional advantage of yielding an upper bound on the Hausdorff dimension of sets of large Weyl sums. Among other technical innov…

Small values of Weyl sums

We prove that the set of $(x_1, \ldots, x_d)\in [0,1)^d$, such that $$ \underline{\lim}_{N\to \infty}\left| \sum_{n=1}^N\exp(2 πi (x_1n+\ldots + x_dn^d)) \right| =0, $$ contains a dense $\mathcal{G}_δ$ set in $[0,1)^d$ and has a positive Hausdorff dimension. Similar statements are also established for the generalised Gaussian sums $$ \sum_{n=1}^N\exp(2πi x n^d), \qquad x \in [0,1). $$

An $L^4$ maximal estimate for quadratic Weyl sums

We show that $$\bigg\|\sup_{0 < t < 1} \big|\sum_{n=1}^{N} e^{2πi (n(\cdot) + n^2 t)}\big| \bigg\|_{L^{4}([0,1])} \leq C_ε N^{3/4 + ε}$$ and discuss some applications to the theory of large values of Weyl sums. This estimate is sharp for quadratic Weyl sums, up to the loss of $N^ε$.

Approximation by Fourier sums in classes of Weyl--Nagy differentiable functions with high exponent of smoothness

We establish asymptotic estimates for the least upper bounds of approximations in the uniform metric by Fourier sums of order $n-1$ of classes of $2π$-periodic Weyl--Nagy differentiable functions, $W^r_{β,p}, 1\le p\le \infty, β\in\mathbb{R},$ for high exponents of smoothness $r\ (r-1\ge \sqrt{n})$. We obtain similar estimates in metrics of the spaces $L_p, 1\le p\le\infty,$ for functional classes $W^r_{β,1}$.

Parametric binomial sums involving harmonic numbers

We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for $p=0,1,2$ and $|t|\leq1$. $$ \sum_{k=1}^{\infty}\frac{H_{k-1}t^k}{k^p\binom{n+k}{k}}\quad \mbox{and}\quad \sum_{k=1}^{\infty}\frac{t^k}{k^p\binom{n+k}{k}}. $$ We also generalize the following relation between the Stirling numbers of the first kind and the Riemann zeta function to polygamma function and give some applications. $$ ζ(n+1)=\sum_{k=n}^{\infty}\frac{s(k,n)}{kk!}, \quad n=1,2,3,... . $$ As examples, \begin{equation*} ζ(3)=\frac{1}{7}\sum_{k=1}^{\infty}\frac{H_{k-1}4^k}{k^2…

Weyl-Nagy类 $W^r_{β,1}$ 上 Fourier 和偏差的估计

Estimates of deviations of Fourier sums on Weyl-Nagy classes $W^r_{β,1}$

本文在Weyl-Nagy类上建立了部分Fourier和偏差的精确上界估计，并由此得到参数n与r以任意关系趋于无穷时的渐近等式。

On $L^{12}$ square root cancellation for exponential sums associated with nondegenerate curves in ${\mathbb R}^4$

We prove sharp $L^{12}$ estimates for exponential sums associated with nondegenerate curves in ${\mathbb R}^4$. We place Bourgain's progress on the Lindel"of hypothesis in a larger framework that contains a continuum of estimates of different flavor. We enlarge the spectrum of methods by combining decoupling with quadratic Weyl sum estimates, to address new cases of interest. All results are proved in the general framework of real analytic curves.

Sufficient conditions for convergence of multiple Fourier series with $J_k$-lacunary sequence of rectangular partial sums in terms of Weyl multipliers

We obtain sufficient conditions for convergence (almost everywhere) of multiple trigonometric Fourier series of functions $f$ in $L_2$ in terms of Weyl multipliers. We consider the case where rectangular partial sums of Fourier series $S_n(x;f)$ have indices $n=(n_1,\dots,n_N) \in \mathbb Z^N$, $N\ge 3$, in which $k$ $(1\leq k\leq N-2)$ components on the places $\{j_1,\dots,j_k\}=J_k \subset \{1,\dots,N\} = M$ are elements of (single) lacunary sequences (i.e., we consider the, so called, multiple Fourier series with $J_k$-lacunary sequence of partial sums). We prove that for any sample $J_k\su…

Parametric Euler Sum Identities

We consider some parametrized classes of multiple sums first studied by Euler. Identities between meromorphic functions of one or more variables generate reduction formulae for these sums.

Weyl交错式的根估计

Rootwise estimates for Weyl alternants

针对Weyl交错式建立根向振荡抵消界，推广至混合导数，并应用于球函数与紧群特征标，恢复并细化已知逐点界。

关于幂和商的上确界

On the supremum of a quotient of power sums

本文研究了幂和商函数的上确界问题，并将其应用于数学金融中的价格影响模型。

Superpositions of coherent states determined by Gauss sums

We describe a family of quantum states of the Schrödinger cat type as superpositions of the harmonic oscillator coherent states with coefficients defined by the quadratic Gauss sums. These states emerge as eigenfunctions of the lowering operators obtained after canonical transformations of the Heisenberg-Weyl algebra associated with the ordinary and fractional Fourier transformation. The first member of this family is given by the well known Yurke-Stoler coherent state.