Alex Barron
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 $N^ε$.
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We obtain new estimates - both upper and lower bounds - on the mean values of the Weyl sums over a small box inside of the unit torus. In particular, we refine recent conjectures of C. Demeter and B. Langowski (2022), and improve some of their results.
We use decoupling theory to prove a sharp (up to $N^ε$ losses) estimate for Vinogradov's mean value theorem in two dimensions
We use decoupling theory to estimate the number of solutions for quadratic and cubic Parsell--Vinogradov systems in two dimensions.
We study an apparently new question about the behaviour of Weyl sums on a subset $\mathcal{X}\subseteq [0,1)^d$ with a natural measure $μ$ on $\mathcal{X}$. For certain measure spaces $(\mathcal{X}, μ)$ we obtain non-trivial bounds for the mean values of the Weyl sums, and for $μ$-almost all points of $\mathcal{X}$ the Weyl sums satisfy the square root cancellation law. Moreover we characterise the size of the exceptional sets in terms of Hausdorff dimension. Finally, we derive variants of the Vinogradov mean value theorem averaging over measure spaces $(\mathcal{X}, μ)$. We obtain general res…
We obtain the exact value of the Hausdorff dimension of the set of coefficients of Gauss sums which for a given $α\in (1/2,1)$ achieve the order at least $N^α$ for infinitely many sum lengths $N$. For Weyl sums with polynomials of degree $d\ge 3$ we obtain a new upper bound on the Hausdorff dimension of the set of polynomial coefficients corresponding to large values of Weyl sums. Our methods also work for monomial sums, match the previously known lower bounds, just giving exact value for the corresponding Hausdorff dimension when $α$ is close to $1$. We also obtain a nearly tight bound in a s…
The authors have recently obtained a lower bound of the Hausdorff dimension of the sets of vectors $(x_1, \ldots, x_d)\in [0,1)^d$ with large Weyl sums, namely of vectors for which $$ \left| \sum_{n=1}^{N}\exp(2πi (x_1 n+\ldots +x_d n^{d})) \right| \ge N^α $$ for infinitely many integers $N \ge 1$. Here we obtain an upper bound for the Hausdorff dimension of these exceptional sets.
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…
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…
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 $$ \sum_{n=1}^N\exp(2πi x n^d), \qquad x \in [0,1). $$
We obtain a new bound on Weyl sums with degree $k\ge 2$ polynomials of the form $(τx+c) ω(n)+xn$, $n=1, 2, \ldots$, with fixed $ω(T) \in \mathbb{Z}[T]$ and $τ\in \mathbb{R}$, which holds for almost all $c\in [0,1)$ and all $x\in [0,1)$. We improve and generalise some recent results of M.~B.~Erdogan and G.~Shakan (2019), whose work also shows links between this question and some classical partial differential equations. We extend this to more general settings of families of polynomials $xn+y ω(n)$ for all $(x,y)\in [0,1)^2$ with $f(x,y)=z$ for a set of $z \in [0,1)$ of full Lebesgue measure, pr…
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 $W^r_{β,1}$.
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。
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…
In this paper we investigate problems on almost everywhere convergence of subsequences of Riemann sums \md0 R_nf(x)=\frac{1}{n}\sum_{k=0}^{n-1}f\bigg(x+\frac{k}{n}\bigg),\quad x\in \ZT. \emd We establish a relevant connection between Riemann and ordinary maximal functions, which allows to use techniques and results of the theory of differentiations of integrals in $\ZR^n$ in mentioned problems. In particular, we prove that for a definite sequence of infinite dimension $n_k$ Riemann sums $R_{n_k}f(x)$ converge almost everywhere for any $f\in L^p$ with $p>1$.
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.
Let $\mathbf{x}=(x_1,\dots,x_n)$ be an $n$-tuple of positive real numbers and the sequence $(x_i)_{i\in\mathbb{Z}}$ be its $n$-periodic extension. Given an $n$-tuple $\mathbf{r}=(r_1,\dots,r_n)$ of positive integers, let $a_i$ be the arithmetic mean of $x_{i+1},\dots,x_{i+r_i}$. We form the cyclic sums $S_n(\mathbf{x},\mathbf{r})=\sum_{i=1}^n x_i/a_{i}$, following the pattern of the long studied Shapiro sums, which correspond to all $r_i=2$, and more general Diananda sums, where all $r_i$ are equal. We find the asymptotics of the $\mathbf{r}$-independent lower bounds $A_{n,*}=\inf_{\mathbf{r}}…
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。
The main aim of this paper is to investigate weighted maximal operators of partial sums of Vilenkin-Fourier series. We also use our results to prove approximation and strong convergence theorems on the martingale Hardy spaces $H_{p},$ when $0<p\leq 1.$
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