Related papers: Bounds for the Cubic Weyl Sum

Assuming some pointwise estimates on certain Weyl's sum, we prove the sharp estimates of the mean value associated to the following exponential sum $$ \sum_{n=1}^N e^{2\pi i tn^d +2\pi i xn}\,. $$

We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of $\alpha n^k$. In particular, when $k\ge 6$ and $\rho(k)$ is defined via the relation…

Let $\lambda_i (n)$ $i= 1, 2, 3$ denote the normalised Fourier coefficients of holomorphic eigenform or Maass cusp form. In this paper we shall consider the sum: \[ S:= \frac{1}{H}\sum_{h\leq H} V\left( \frac{h}{H}\right)\sum_{n\leq N}…

We provide a new exponent for the Sum-Product conjecture on $\mathbb{R} $. Namely for $A \subset \mathbb{R}$ finite, \[ \max \left\{ \left\lvert A+A \right\rvert , \left\lvert AA \right\rvert \right\} \gg_{\epsilon} \left\lvert A…

Let $\mathcal{A}$ be a finite subset of $\mathbb{N}$ including $0$ and $f_\mathcal{A}(n)$ be the number of ways to write $n=\sum_{i=0}^{\infty}\epsilon_i2^i$, where $\epsilon_i\in\mathcal{A}$. We consider asymptotics of the summatory…

Upper bound estimates for the exponential sum $$ \sum_{K<\kappa_j\le K'<2K} \alpha_j H_j^3(1/2) \cos(\k_j\log({4{\rm e}T\over \kappa_j})) \qquad(T^\epsilon \le K \le T^{1/2-\epsilon}) $$ are considered, where $\alpha_j =…

Let $n$ be a cubefree natural number and $p\geq 5$ be a prime number. Assume that $n$ is not expressible as a sum of the form $x^3+y^3$, where $x,y\in \mathbb{Q}$. In this note, we study the solutions (or lack thereof) to the equation…

We obtain $\Omega$-results for linear exponential sums with rational additive twists of small prime denominators weighted by Hecke eigenvalues of Maass cusp forms for the group $\mathrm{SL}_3(\mathbb Z)$. In particular, our $\Omega$-results…

A symmetrized lattice of $2n$ points in terms of an irrational real number $\alpha$ is considered in the unit square, as in the theorem of Davenport. If $\alpha$ is a quadratic irrational, the square of the $L^2$ discrepancy is found to be…

We give a new bound on the number of collinear triples for two arbitrary subsets of a finite field. This improves on existing results which rely on the Cauchy inequality. We then us this to provide a new bound on trilinear and quadrilinear…

Let $A\subset\mathbb{N}$, $\alpha\in(0,1)$, and for $x\in\mathbb{R}$ let $e(x):=e^{2\pi ix}$. We set $$S_{A}(\alpha,N):=\sum_{\substack{n\in A\n\leq N}}e(n\alpha).$$ Recently, Lambert A'Campo proposed the following question: is there an…

We evaluate the convolution sum $\displaystyle W_{a,b}(n):= \sum_{al+bm=n} \hspace{-3mm} \sigma(l) \sigma(m)$ for $(a,b)=(1,28), (4,7), (2,7)$ for all positive integers $n$. We use a modular form approach. We also re-evaluate the known sums…

We have derived explicit expressions for the sum rules of order one of the eigenvalues of the negative Laplacian on two dimensional domains of arbitrary shape. Taking into account the leading asymptotic behavior of these eigenvalues, as…

We prove that the dimension of the space of primitive Vassiliev invariants of degree n grows - as n tends to infinity - faster than Exp(c Sqrt(n)) for any c < Pi Sqrt (2/3). The proof relies on the use of the weight systems coming from the…

Let $N(X)$ be the number of integral zeros $(x_1,\dots,x_6)\in [-X,X]^6$ of $\sum_{1\le i\le 6} x_i^3$. Works of Hooley and Heath-Brown imply $N(X)\ll_\epsilon X^{3+\epsilon}$, if one assumes automorphy and GRH for certain Hasse--Weil…

We construct a non - improved exponential bounds for distribution of normed sums of i.,i.d. random variables with random numbers of summand.

Let $$S(N) = \sum_{n \sim N}^{\text{smooth}} \, \lambda_{f}(n) \, \chi(n),$$ where $\lambda_{f}(n)$'s are Fourier coefficients of Hecke-eigen form, and $\chi$ is a primitive character of conductor $p^{r}$. In this article we prove a…

We introduce a general class $F_0$ of additive functions $f$ such that $f(p) = 1$ and prove a tight bound for exponential sums of the form $\sum_{n \le x} f(n) e(\alpha n)$ where $f \in F_0$ and $e(\theta) = \exp(2\pi i \theta)$. Both…

We give an upper bound for the exponential sum over squarefree integers. This establishes a conjecture by Br\"udern and Perelli.

Let $C=Z(f)$ be a reduced plane curve of degree $6k$, with only nodes and ordinary cusps as singularities. Let $I$ be the ideal of the points where $C$ has a cusp. Let $\oplus S(-b_i)\to \oplus S(-a_i) \to S\to S/I$ be a minimal resolution…