Related papers: Perturbations of Weyl sums

The Franel numbers given by $f_n=\sum_{k=0}^n\binom{n}{k}^3$ ($n=0,1,2,\ldots$) play important roles in both combinatorics and number theory. In this paper we initiate the systematic investigation of fundamental congruences for the Franel…

We prove several supercongruences involving the harmonic number of order two $H_n^{(2)}:=\sum_{k=1}^n1/k^2$. For example, if $p>5$ is prime and $\alpha$ is $p$-integral, then we can completely determine $$…

To begin, we find certain formulas $Q(k,\alpha)= G_1^k(\alpha) G_2^k(\alpha)$, for $k = -1, 0, 1,...,9$. These yield that part of the total separability probability, $P(k,\alpha)$, for generalized (real, complex, quaternionic,\ldots)…

Let $H_k$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\geq 2$ for the full modular group $SL(2, \mathbb{Z})$, and let $j\geq 3$ be any fixed integer. For $f\in H_k$, we write $\lambda_{{\rm{sym}^j…

In this paper we obtain a series of asymptotic formulae in the sum--product phenomena over the prime field $\mathbf{F}_p$. In the proofs we use usual incidence theorems in $\mathbf{F}_p$, as well as the growth result in ${\rm SL}_2…

The Delannoy polynomial $D_n(x)$ is defined by $$ D_n(x)=\sum_{k=0}^{n}{n\choose k}{n+k\choose k}x^k. $$ We prove that, if $x$ is an integer and $p$ is a prime not dividing $x(x+1)$, then \begin{align*} \sum_{k=0}^{p-1}(2k+1)D_k(x)^3…

Let $k$ be a natural number. We consider $k$-times continuously-differentiable real-valued functions $f:E\to\mathbb{R}$, where $E$ is some interval on the line having positive length. For $0<\alpha<1$ let $I_\alpha(f)$ denote the set of…

Alon, Shpilka and Umans considered the following version of usual sunflower-free subset: a subset $\mbox{$\cal F$}\subseteq \{1,\ldots ,D\}^n$ for $D>2$ is sunflower-free if for every distinct triple $x,y,z\in \mbox{$\cal F$}$ there exists…

Let $c$ be an element of the Weyl algebra $W(d)$ which is given by a strictly positive operator in the Schr"odinger representation. It is shown that, under some conditions, there exist elements $b_1,...,b_d$ in $W(d)$ such that $b_1 c b_1^*…

Let $n,s,k$ be three positive integers such that $1\leq s\leq(n-k+1)/k$ and let $[n]=\{1,\ldots,n\}$. Let $H$ be a $k$-graph with vertex set $\{1,\ldots,n\}$, and let $e(H)$ denote the number of edges of $H$. Let $\nu(H)$ and $\tau(H)$…

Let $A,B$ be nonempty subsets of a an abelian group $G$. Let $N_i(A,B)$ denote the set of elements of $G$ having $i$ distinct decompositions as a product of an element of $A$ and an element of $B$. We prove that $$ \sum _{1\le i \le t} |N_i…

We stratify the $\mathrm{SL}_3$ big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the $\mathrm{SL}_3$ long word Kloosterman sum is decomposed into…

The main results of this paper concern growth in sums of a $k$-convex function $f$. Firstly, we streamline the proof of a growth result for $f(A)$ where $A$ has small additive doubling, and improve the bound by removing logarithmic factors.…

Let $\{a_{1}, a_{2},\ldots, a_{n},\ldots\}$ be a sequence of complex numbers which has at most polynomial growth and satisfies an extra assumption. In this paper, inspired by a recent work of Sasane, we give an explanation of the sum…

For a periodic function $f$ with bounded variation and integral zero on its period interval, we show that $\sum_{k=1}^\infty c_k^2 (\log\log k)^\gamma <\infty$, $\gamma>4$ implies the almost everywhere convergence of $\sum_{k=1}^\infty c_k…

Let $\mathrm{R}$ be a real closed field and $Q_1, \ldots, Q_{\ell} \in \mathrm{R}[X_1, \ldots,X_k]$ such that for each $i, 1 \leq i \leq \ell$, $\mathrm{deg} (Q_i) \leq d_i$. For $1 \leq i \leq \ell$, denote by $\mathcal{Q}_i = \{Q_1,…

For all nonnegative integers n, the Franel numbers are defined as $$ f_n=\sum_{k=0}^n {n\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0 \pmod{2n^2},…

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. Let $A(n)$…

In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and for all $f,g$ convex or concave functions, we have $$|A + B|^{38}|f(A) + g(B)|^{38}…

TO BE PUBLISHED BY ISRAEL JOURNAL OF MATHEMATICS. We study the mean $\sum_{x\in\mathcal{X}} \bigl|\sum_{p\le N}{}u_p e(xp)\bigr|^{\ell}$ when $\ell$ covers the full range $[2,\infty)$ and $\mathcal{X}\subset\mathbb{R}/\mathbb{Z}$ is a…