arXiv subjects

Jack Anderson

Publications and source records attributed to Jack Anderson.

6 recordsLinked to original sources

arXiv2025

On denominators of consecutive $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions

The sequence $({\mathscr S}_Q)_Q$ of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions was defined in our previous work by ${\mathscr S}_Q := \{ a/q \in {\mathbb Q} \cap (0,1]: q+a+\bar{a} \le Q\}$, where $\bar{a}$ is the multiplicative inverse of $a\pmod{q}$ in $[1,q)$. Here, we prove that the set of $Q$-scaled denominators of consecutive fractions in ${\mathscr S}_Q$ is dense in the region ${\mathcal V}:=\{ (x,y)\in [0,1]^2 : \max \{ (1-3x)/2,2x-1\} \le y \le \max \{ x,1-x\} \}$, and provide a formula for their distribution in ${\mathcal V}$ as $Q\rightarrow \infty$.

math.NT↗

arXiv2025

On the distribution of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions

We consider the set ${\mathscr S}_Q$ of Farey fractions $d/b$ of order $Q$ with the property that there exists a matrix $\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \operatorname{SL}(2,{\mathbb Z})$ of trace at most $Q$, with positive entries and $a\ge \max\{ b,c\}$. For every $Q\ge 3$, the set ${\mathscr S}_Q \cup \{ 0\}$ is shown to define a unimodular partition of the interval $[0,1]$. We also prove that the elements of ${\mathscr S}_Q$ are asymptotically distributed with respect to the probability measure with density $(1/(1+x) -1/(2+x) )/\log (4/3) $ and that the sequence of sets $({\mathscr S}_Q)_Q$ has a limiting gap distribution as $Q\rightarrow \infty$.

math.NT↗

arXiv2024

Arithmetic Polygons and Sums of Consecutive Squares

We introduce and study arithmetic polygons. We show that these arithmetic polygons are connected to triples of square pyramidal numbers. For every odd $N\geq3$, we prove that there is at least one arithmetic polygon with $N$ sides. We also show that there are infinitely many arithmetic polygons with an even number of sides.

math.NT↗

arXiv2023

Angular distribution towards the points of the neighbor-flips modular curve seen by a fast moving observer

Let $h$ be a fixed non-zero integer. For every $t\in \mathbb{R}_+$ and every prime $p$, consider the angles between rays from an observer located at the point $(-tJ_p^2,0)$ on the real axis towards the set of all integral solutions $(x,y)$ of the equation $y^{-1}-x^{-1}\equiv h \pmod{p}$ in the square $[-J_p,J_p]^2$, where $J_p=(p-1)/2$. We prove the existence of the limiting gap distribution for this set of angles as $p\rightarrow \infty$, providing explicit formulas for the corresponding density function, which turns out to be independent of $h$.

math.NT↗

arXiv2023

Counterintuitive patterns on angles and distances between lattice points in high dimensional hypercubes

Let $\mathcal{S}$ be a finite set of integer points in $\mathbb{R}^d$, which we assume has many symmetries, and let $P\in\mathbb{R}^d$ be a fixed point. We calculate the distances from $P$ to the points in $\mathcal{S}$ and compare the results. In some of the most common cases, we find that they lead to unexpected conclusions if the dimension is sufficiently large. For example, if $\mathcal{S}$ is the set of vertices of a hypercube in $\mathbb{R}^d$ and $P$ is any point inside, then almost all triangles $PAB$ with $A,B\in\mathcal{S}$ are almost equilateral. Or, if $P$ is close to the center of the cube, then almost all triangles $PAB$ with $A\in \mathcal{S}$ and $B$ anywhere in the hypercube are almost right triangles.

math.CO↗

arXiv2023

Distribution of angles to lattice points seen from a fast moving observer

We consider a square expanding with constant speed seen from an observer moving away with constant acceleration and study the distribution of angles between rays from the observer towards the lattice points in the square. We prove the existence of the gap distribution as time tends to infinity and provide explicit formulas for the corresponding density function.

math.NT↗