Related papers: Sums of three cubes, II

We provide an upper bound on the uniform exponent of approximation to a triple (xi, xi^2, xi^3) by rational numbers with the same denominator, valid for any transcendental real number xi. This upper bound refines a previous result of…

We establish estimates for linear correlation sums involving sums of three positive integral cubes. Under appropriate conditions, the underlying methods permit us to establish the solubility of systems of homogeneous linear equations in…

We consider large values of long linear exponential sums involving Fourier coefficients of holomorphic cusp forms. The sums we consider involve rational linear twists $e(nh/k)$ with sufficiently small denominators. We prove both pointwise…

We establish that almost every positive integer $n$ is the sum of four cubes, two of which are at most $n^{\theta}$, as long as $\theta\geq192/869$. An asymptotic formula for the number of such representations is established when…

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weyl sums. We focus attention on the problem of representing large…

We examine a family of three-dimensional exponential sums with monomials and provide estimates which are in some instances sharper than those stemming from approaches entailing the use of existing bounds pertaining to analogous sums.

Higher-dimensional Dedekind sums are defined as a generalization of a recent 1-dimensional probability model of Dilcher and Girstmair to a d-dimensional cube. The analysis of the frequency distribution of marked lattice points leads to new…

Let $f(x, y) \in \mathbb{Z}[x, y]$ be a cubic form with non-zero discriminant, and for each integer $m \in \mathbb{Z}$, let, $N_{f}(m)=\#\left\{(x, y) \in \mathbb{Z}^{2}: f(x, y)=m\right\} $. In 1983, Silverman proved that…

In this paper, we obtain the maximal estimate for the Weyl sums on the torus $\mathbb{T}^d$ with $d\geq 2$, which is sharp up to the endpoint. We also consider two variants of this problem which include the maximal estimate along the…

We investigate a version of Waring's Problem over quaternion rings, focusing on cubes in quaternion rings with integer coefficients. We determine the global upper and lower bounds for the number of cubes necessary to represent all such…

There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these…

We establish a lower bound for the frequency with which an irreducible monic cubic polynomial with negative discriminant can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by…

We obtain asymptotic formulas for the $2k$th moments of partially smoothed divisor sums of the M\"obius function. When $2k$ is small compared with $A$, the level of smoothing, then the main contribution to the moments come from integers…

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…

Recently, Alp\"oge-Bhargava-Shnidman determined the average size of the $2$-Selmer group in the cubic twist family of any elliptic curve over $\mathbb{Q}$ with $j$-invariant $0$. We obtain the distribution of the $3$-Selmer groups in the…

We show that $$\bigg\|\sup_{0 < t < 1} \big|\sum_{n=1}^{N} e^{2\pi i (n(\cdot) + n^2 t)}\big| \bigg\|_{L^{4}([0,1])} \leq C_{\epsilon} N^{3/4 + \epsilon}$$ and discuss some applications to the theory of large values of Weyl sums. This…

In this paper, we establish a new estimate (including lower and upper bounds) for an important quantity involved in the convergence analysis of smoothed aggregation algebraic multigrid methods. The new upper bound improves the existing…

The main goal of this expository article is to survey recent progress on the arithmetic Siegel-Weil formula and its applications. We begin with the classical sum of two squares problem and put it in the context of the Siegel-Weil formula.…

We give an asymptotic for the number of prime solutions to $Q(x_1,\dots, x_8) = N$, subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$. The argument initially proceeds via the circle method, but this does not…

In this paper, we prove the smooth cubic moments vanish for the Hecke--Maass cusp forms, which gives a new case of the random wave conjecture. In fact, we can prove a polynomial decay for the smooth cubic moments, while for the smooth…