Exact Exponent of Remainder Term of Gelfond's Digit Theorem in Binary Case

Ranks of Quotients, Remainders and $p$-Adic Digits of Matrices

For a prime $p$ and a matrix $A \in \mathbb{Z}^{n \times n}$, write $A$ as $A = p (A \,\mathrm{quo}\, p) + (A \,\mathrm{rem}\, p)$ where the remainder and quotient operations are applied element-wise. Write the $p$-adic expansion of $A$ as $A = A^{[0]} + p A^{[1]} + p^2 A^{[2]} + \cdots$ where each $A^{[i]} \in \mathbb{Z}^{n \times n}$ has entries between $[0, p-1]$. Upper bounds are proven for the $\mathbb{Z}$-ranks of $A \,\mathrm{rem}\, p$, and $A \,\mathrm{quo}\, p$. Also, upper bounds are proven for the $\mathbb{Z}/p\mathbb{Z}$-rank of $A^{[i]}$ for all $i \ge 0$ when $p = 2$, and a conje…

Number systems and the Chinese Remainder Theorem

A well-known generalisation of positional numeration systems is the case where the base is the residue class of $x$ modulo a given polynomial $f(x)$ with coefficients in (for example) the integers, and where we try to construct finite expansions for all residue classes modulo $f(x)$, using a suitably chosen digit set. We give precise conditions under which direct or fibred products of two such polynomial number systems are again of the same form. The main tool is a general form of the Chinese Remainder Theorem. We give applications to simultaneous number systems in the integers.

短算术项表示有限域上魏尔斯特拉斯标准形式椭圆曲线的基数

Short arithmetic terms for the cardinality of elliptic curves over rings of remainder classes and over finite fields

该研究改进了椭圆曲线计数的算术项构造方法，降低了中间整数位数，对素模p≥17时大幅减少了运算次数，提升了有限域上魏尔斯特拉斯标准形式椭圆曲线基数的计算效率。

基 $b(T)$ 下数字受限的不可约多项式

Irreducible polynomials with restricted digits in base $b(T)$

本文研究有限域上多项式环中数字受限的不可约多项式计数，利用傅里叶分析得到渐近公式，并推广圆法至任意多项式基。