Related papers: Pollard's theorem in general abelian groups

We prove a probabilistic Fourier extension theorem that says Fourier extension holds when averaged over certain smooth Alpert multipliers. The proofs use smooth Alpert wavelets with the classical techniques of stationary phase and…

We show that various identities from [1] and [3] involving Gould-Hopper polynomials can be deduced from the real but also complex orthogonal invariance of multivariate Gaussian distributions. We also deduce from this principle a useful…

In this note it is established that, for any finite set $A$ of real numbers, there exist two elements $a,b \in A$ such that $$|(a+A)(b+A)| \gg \frac{|A|^2}{\log |A|}.$$ In particular, it follows that $|(A+A)(A+A)| \gg \frac{|A|^2}{\log…

This paper introduces the concept of slender generalized groups, extending the classical notion of slender abelian groups to the setting of generalized groups (completely simple semigroups). We establish fundamental properties of slender…

When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume…

We show that a principally polarized abelian variety over a field $k$ is, as an abelian variety, a direct summand of a product of Jacobians of curves which contain a $k$-point if and only if the polarization and the minimal class are both…

We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A qualitative version of this generalisation was…

A finite sampling theory associated with a unitary representation of a finite non Abelian group $\mathbf{G}$ on a Hilbert space is stablished. The non Abelian group $\mathbf{G}$ is a knit product $\mathbf{N}\bowtie \mathbf{H}$ of two finite…

In this paper we study a group theoretical generalization of the well-known Gauss's formula that uses the generalized Euler's totient function introduced in [11].

The author in [7] was proved the generalized remainder and quotient theorems of polynomial in one indeterminate where the divisor is complete factorization to linear factors. In this paper we give the formula for the generalized remainder…

We say that an abelian variety $A_{/\mathbf Q}$ of dimension $g$ is {\em prosaic} if it is semistable, with good reduction at 2 and its points of order $2$ generate a $2$-extension of ${\mathbf Q}$. For $p \equiv 1 \bmod{8}$, let $M_u$ be…

We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group $G$ and an $n$-element subset $Y \subseteq G$ we show that if $m \ll s^2/(\log n)^2$, then the number of subsets $A \subseteq Y$ with $|A| = s$ and…

Let $G$ be a finite group and $\varphi(G)=|\{a\in G \mid o(a)=\exp(G)\}|$, where $o(a)$ denotes the order of $a$ in $G$ and $\exp(G)$ denotes the exponent of $G$. Under a natural hypothesis, in this note we determine the groups $G$ such…

Let $\alpha,\beta \in \mathbb{R}_{>0}$ be such that $\alpha,\beta$ are quadratic and $\mathbb{Q}(\alpha)\neq \mathbb{Q}(\beta)$. Then every subset of $\mathbb{R}^n$ definable in both $(\mathbb{R},{<},+,\mathbb{Z},x\mapsto \alpha x)$ and…

Let $A$ be a finite dimensional algebra of finite global dimension over a finite field. In the present paper, we introduce certain elements in Bridgeland's Hall algebra of $A$, and give a multiplication theorem of these elements. In…

The solubilizer of an element $x$ of a profinite group $G$ is the set of the elements $y$ of $G$ such that the subgroup of $G$ generated by $x$ and $y$ is prosoluble. We propose the following conjecture: the solubilizer of $x$ in $G$ has…

The twisted $T$-adic exponential sum associated to $x^{d}+\lambda x$ is studied. If $\lambda\neq0,$ then an explicit arithmetic polygon is proved to be the Newton polygon of the twisted $C$-function of the T-adic exponential sum. It gives…

Let $G$ be a finite abelian group, let $0 < \alpha < 1$, and let $A \subseteq G$ be a random set of size $|G|^\alpha$. We let $$ \mu(A) = \max_{B,C:|B|=|C|=|A|}|\{(a,b,c) \in A \times B \times C : a = b + c \}|. $$ The issue is to determine…

For each natural number $m\ge 3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. In this paper, with the help of the congruence theta function, we establish conditions on $a$, $b$,…

We studies the Newton polygon for the L-function of toric exponential sums attached to a family of two variable generalized hyperkloosterman sum,$f_{t}(x,y)=x^{n}+y+\frac{t}{xy}$ with $t$ the parameter. The explicit Newton polygon is…