An analog of the arithmetic triangle obtained by replacing the products by the least common multiples

The Maximum Number of Three Term Arithmetic Progressions, and Triangles in Cayley Graphs

Let $G$ be a finite Abelian group. For a subset $S \subseteq G$, let $T_3(S)$ denote the number of length three arithemtic progressions in $S$ and Prob[$S$] $= \frac{1}{|S|^2}\sum_{x,y \in S} 1_S(x+y)$. For any $q \ge 1$ and $α\in [0,1]$, and any $S \subseteq G$ with $|S| = \frac{|G|}{q+α}$, we show $\frac{T_3(S)}{|S|^2}$ and Prob[$S$] are bounded above by $\max\left(\frac{q^2-αq+α^2}{q^2},\frac{q^2+2αq+4α^2-6α+3}{(q+1)^2},γ_0\right)$, where $γ_0 < 1$ is an absolute constant. As a consequence, we verify a graph theoretic conjecture of Gan, Loh, and Sudakov for Cayley graphs.

Metric and arithmetic properties of mediant-Rosen maps

A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued fractions is sufficient to determine the values that Haas and Series found by hyperbolic geometry.

On arithmetic properties of periods for some rational differential forms over $\mathbb{Q}$ on the Fermat curve $F_2$ of degree 2

In this paper, we will define analogues of multiple zeta values by replacing the differential forms defining multiple zeta values with some $\mathbb{Q}$-rational differential forms on the Fermat curve $F_2$ of degree 2 and discuss their arithmetic properties. We also investigate a motivic structure of the motivic periods corresponding to our periods. However, in order to study them, the current theory for motivic zeta elements is insufficient, and it leads us to study the base extension of the space of the motivic periods $\mathcal{H}_4$ of level 4 and its Galois invariant part.

一个算术Yau-Zaslow公式

An arithmetic Yau-Zaslow formula

本文提出一个算术 refinement 的Yau-Zaslow公式，通过引入motivic欧拉特征，推导出其他相关不变量的公式，包括实K3表面实有理曲线计数和上同调行列式。

一些涉及算术函数与 Bhargava 阶乘的丢番图方程

Some Diophantine equations involving arithmetic functions and Bhargava factorials

本文将 Luca 关于阶乘与 Euler 函数、除数和及除数计数函数的丢番图方程有限解结论，推广到多阶乘乘积和除数 k 次幂和函数，并扩展至 Bhargava 阶乘。

Partition-theoretic formulas for arithmetic densities, II

In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers $\mathbb N$ as limiting values of $q$-series as $q\to ζ$ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of $\mathbb N$ by analogous structures in the integer partitions $\mathcal P$. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of…

Shimura curves for level-3 subgroups of the (2,3,7) triangle group, and some other examples

We determine the Shimura modular curve X_0(3) and the Jacobian of the Shimura modular curve X_1(3) associated with the congruence subgroups Gamma_0(3), Gamma_1(3) of the (2,3,7) triangle group. This group is known to be arithmetic and associated with a quaternion algebra A/K ramified at two of the three real places of K=Q(\cos 2π/7) and at no finite primes of K. Since the rational prime 3 is inert in K, the covering X_0(3)/X(1) has degree 28 and Galois group PSL_2(F_27). We determine X_0(3) by computing this cover. We find that X_0(3) is an elliptic curve of conductor 147=3*7^2 over Q, as is t…