A variant of Waring's Problem for the ring of integers modulo n

幂模的大筛法不等式与华林问题

Large sieve inequalities with power moduli and Waring's problem

本文通过将大筛法不等式与华林问题的受限变体关联，并利用经典除数界和 Wooley 的指数和均值结果，改进了所有 k≥4 情形下带 k 次幂模的大筛法不等式，同时给出了基于 Hardy-Littlewood 猜想公式的条件性结果。

短区间内模$p$的CM椭圆曲线的无平方因子阶

Square-free orders for CM elliptic curves modulo $p$ in short intervals

本文研究有理数域上CM椭圆曲线模素数的无平方因子阶计数问题，改进了已有渐近公式，得到无条件短区间渐近结果，且适用区间比广义黎曼假设下的更短，同时优化了CM椭圆曲线循环性问题的短区间变体结论。

A variant of Davenport's constant

Let p be a prime number. Let G be a finite abelian p-group of exponent n (written additively) and A be a non-empty subset of $]n[:= \{1,2,..., n\}$ such that elements of A are incongruent modulo p and non-zero modulo p. Let $k \geq D(G)/|A|$ be any integer where D(G) denotes the well-known Davenport's constant. In this article, we prove that for any sequence g_1, g_2, ..., g_k (not necessarily distinct) in G, one can always extract a subsequence g_{i_1}, g_{i_2}, ..., g_{i_\ell} with $1\leq \ell \leq k$ such that \begin{equation*} \sum_{j=1}^\ell a_{j}g_{i_j} = 0 {in} G, \end{equation*} where …

Points on polynomial curves in small boxes modulo an integer

Given an integer $q$ and a polynomial $f\in \mathbb Z_{q}[X]$ of degree $d$ with coefficients in the residue ring $\mathbb Z_q=\mathbb Z/q\mathbb Z,$ we obtain new results concerning the number of solutions to congruences of the form $$y\equiv f(x) \pmod{q},$$ with integer variables lying in some cube $\mathcal B$ of side length $H$. Our argument uses ideas of Cilleruelo, Garaev, Ostafe and Shparlinski which reduces the problem to the Vinogradov mean value theorem and a lattice point counting problem. We treat the lattice point problem differently using transference principles from the Geometr…

Geometric configurations in the ring of integers modulo $p^{\ell}$

We study variants of the Erd\H os distance problem and dot products problem in the setting of the integers modulo $q$, where $q = p^{\ell}$ is a power of an odd prime.

Effective bounds on multiplicatively dependent orbits of integer polynomials modulo S-integers

We obtain effective bounds on the heights of algebraic integers whose orbits contain multiplicatively dependent values modulo S-integers. Our method is based on a new upper bound on the so-called S-height of polynomial values over the ring of integers of $\mathbb{K}$. Our results provide an effective variant of a recent result of A.Bérczes, A.Ostafe, I.E.Shparlinski and J.H.Silverman (arXiv:1811.04971) on multiplicative dependence modulo a finitely generated subgroup by eliminating the use of non-effective results by K.F.Roth and G.Faltings.

具有给定原根的最小素数及大筛法的一个变体

The least prime with a given primitive root and a variant of the larger sieve

本文在广义黎曼猜想下，将给定原根的最小素数上界改进为对数三次方量级，并证明大筛法新变体，优于现有结果。

Primes in arithmetic progressions and nonprimitive roots

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb Z/p\mathbb Z)^*,$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each primitive residue class contains a positive natural density subset of primes $p$ not having $g$ as a $t$-near primitive root and prove a more difficult variant.

On the statistics of the minimal solution of a linear Diophantine equation and uniform distribution of the real part of orbits in hyperbolic spaces

We study a variant of a problem considered by Dinaburg and Sinai on the statistics of the minimal solution to a linear Diophantine equation. We show that the signed ratio between the Euclidean norms of the minimal solution and the coefficient vector is uniformly distributed modulo one. We reduce the problem to an equidistribution theorem of Anton Good concerning the orbits of a point in the upper half-plane under the action of a Fuchsian group.

模环境下的 Erdős 不同子集和问题

The Erdős distinct subset sums problem in a modular setting

本文研究模环境下 Erdős 不同子集和问题，证明当模为 $N=2^n+t$ 时，任意 $n$ 元不同子集和集合的最大元至少约为 $N/3$，并证明该常数最优，同时刻画了小 $t$ 情形下的集合结构。

整数和的循环

Cycles of Sums of Integers

本文研究模整数环上线性映射 $T$ 的周期，将其视为 Ducci 序列的变体，证明模 $m$ 的周期可由 $m$ 的素因子周期推导，并刻画了素数模情形下属于循环的元组。

Linnik's problem in fiber bundles over quadratic homogeneous varieties

We compute the statistics of $SL_{d}(\mathbb{Z})$ matrices lying on level sets of an integral polynomial defined on $SL_{d}(\mathbb{R})$, a result that is a variant of the well known theorem proved by Linnik about the equidistribution of radially projected integral vectors from a large sphere into the unit sphere. Using the above result we generalize the work of Aka, Einsiedler and Shapira in various directions. For example, we compute the joint distribution of the residue classes modulo $q$ and the properly normalized orthogonal lattices of primitive integral vectors lying on the level set $-…