Related papers: Maximal estimates for the Weyl sums on $\mathbb{T}…
We prove two bounds for discrete moments of Weyl sums. The first one can be obtained using a standard approach. The second one involves an observation how this method can be improved, which leads to a sharper bound in certain ranges. The…
We consider quadratic Weyl sums $S_N(x;c,\alpha)=\sum_{n=1}^N\exp\{2\pi i((\frac{1}{2}n^2+cn)x+\alpha n)\}$ for $c=\alpha=0$ (the rational case) or $(c,\alpha)\notin\mathbb{Q}^2$ (the irrational case), where $x$ is randomly distributed…
The main results extend to sums over primes in a short interval earlier estimates by the author for "long" Weyl sums over primes.
Estimates are provided for $s$th moments of cubic smooth Weyl sums, when $4\le s\le 8$, by enhancing the author's iterative method that delivers estimates beyond classical convexity. As a consequence, an improved lower bound is presented…
We establish the submaximal symmetry dimension for Riemannian and Lorentzian conformal structures. The proof is based on enumerating all subalgebras of orthogonal Lie algebras of sufficiently large dimension and verifying if they stabilize…
Lacey and Thiele have recently obtained a new proof of Carleson's theorem on almost everywhere convergence of Fourier series. This paper is a generalization of their techniques (known broadly as time-frequency analysis) to higher…
We study the number of exponentially small singular values of the semiclassical $\overline{\partial}$ operator on exponentially weighted $L^2$ spaces on the two-dimensional torus. Accurate upper and lower bounds on the number of such…
In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due…
Let $q$ be a prime, $P \geq 1$ and let $N_q(P)$ denote the number of rational primes $p \leq P$ that split in the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$. The first part of this paper establishes various unconditional and…
We describe a new method to bound certain higher-dimensional exponential sums which are associated with tori in symplectic groups over finite fields. Our method is based on the self-reducibility property of the Weil representation. As a…
In this thesis, we attempt to gain a more complete insight into Double Layer Theories in Weyl Gravity. In order to do this, we first establish the premise of Weyls Theory, including its provenance, development and flaws. This is all…
We provide estimates for $s^{\rm th}$ moments of biquadratic smooth Weyl sums, when $10\le s\le 12$, by enhancing the second author's iterative method that delivers estimates beyond the classical convexity barrier. As a consequence, all…
Given a truncated perturbation expansion of a physical quantity, one can, under certain circumstances, obtain lower or upper bounds (or both) to the sum of the full perturbation series by using the Borel transform and a variational…
The gauge formulation of Einstein gravity in AdS$_3$ background leads to a boundary theory that breaks modular symmetry and loses the covariant form. We examine the Weyl anomaly for the cylinder and torus manifolds. The divergent term is…
We prove that there exist positive constants $C$ and $c$ such that for any integer $d \ge 2$ the set of ${\mathbf x}\in [0,1)^d$ satisfying $$ cN^{1/2}\le \left|\sum^N_{n=1}\exp\left (2 \pi i \left (x_1n+\ldots+x_d n^d\right)\right)…
In 1965, Bollob\'as proved that for a Bollob\'as set-pair system $\{(A_i,B_i)\mid i\in[m]\}$, the maximum value of $\sum_{i=1}^m\binom{|A_i|+|B_i|}{A_i}^{-1}$ is $1$. Heged\"{u}s and Frankl recently extended the concept of Bollob\'as…
In this paper we study Weyl sums over friable integers (more precisely $y$-friable integers up to $x$ when $y = (\log x)^C$ for a large constant $C$). In particular, we obtain an asymptotic formula for such Weyl sums in major arcs,…
We improve the standard Weyl estimate for quartic exponential sums in which the argument is a quadratic irrational. Specifically we show that \[\sum_{n\le N} e(\alpha n^4)\ll_{\ep,\alpha}N^{5/6+\ep}\] for any $\ep>0$ and any quadratic…
In this paper, we consider a "compensated" random sum that arises from numerical approximation of stochastic integrations and differential equations. We show that the compensated sum exhibits some surprising cancellations among its…
We present a hybrid approach to bounding exponential sums over kth powers via Vinogradov's mean value theorem, and derive estimates of utility for exponents k of intermediate size.