arXiv subjects

David Harry Richman

Publications and source records attributed to David Harry Richman.

6 recordsLinked to original sources

arXiv2024

The family of $a$-floor quotient partial orders

An approximate divisor order is a partial order on the positive integers $\mathbb{N}^+$ that refines the divisor order and is refined by the additive total order. A previous paper studied such a partial order on $\mathbb{N}^+$, produced using the floor function. A positive integer $d$ is a floor quotient of $n$, denoted $d \,\preccurlyeq_{1}\, n$, if there is a positive integer $k$ such that $d = \lfloor{n / k}\rfloor$. The floor quotient relation defines a partial order on the positive integers. This paper studies a family of partial orders, the $a$-floor quotient relations $\,\preccurlyeq_{a}\,$, for $a \in \mathbb{N}^+$, which interpolate between the floor quotient order and the divisor order on $\mathbb{N}^+$. The paper studies the internal structure of these orders.

math.NT↗

arXiv2024

The distribution of Weierstrass points on a tropical curve

We show that on a metric graph of genus $g$, a divisor of degree $n$ generically has $g(n-g+1)$ Weierstrass points. For a sequence of generic divisors on a metric graph whose degrees grow to infinity, we show that the associated Weierstrass points become distributed according to the Zhang canonical measure. In other words, the limiting distribution is determined by effective resistances on the metric graph. This distribution result has an analogue for complex algebraic curves, due to Neeman, and for curves over non-Archimedean fields, due to Amini.

math.AG↗

arXiv2024

Lower rational approximations and Farey staircases

For a real number $x$, call $\frac1n \lfloor nx \rfloor$ the $n$-th lower rational approximation of $x$. We study the functions defined by taking the cumulative average of the first $n$ lower rational approximations of $x$, which we call the Farey staircase functions. This sequence of functions is monotonically increasing. We determine limit behavior of these functions and show that they exhibit fractal structure under appropriate normalization.

math.NT↗

arXiv2023

The floor quotient partial order

A positive integer $d$ is a floor quotient of $n$ if there is a positive integer $k$ such that $d = \lfloor n/k \rfloor$. The floor quotient relation defines a partial order on the positive integers. This paper studies the internal structure of this partial order and its Möbius function.

math.NT↗

arXiv2023

The tropical Manin-Mumford conjecture

In analogy with the Manin-Mumford conjecture for algebraic curves, one may ask how a metric graph under the Abel-Jacobi embedding intersects torsion points of its Jacobian. We show that the number of torsion points is finite for metric graphs of genus $g\geq 2$ which are biconnected and have edge lengths which are "sufficiently irrational" in a precise sense. Under these assumptions, the number of torsion points is bounded by $3g-3$. Next we study bounds on the number of torsion points in the image of higher-degree Abel-Jacobi embeddings, which send $d$-tuples of points to the Jacobian. This motivates the definition of the "independent girth" of a graph, a number which is a sharp upper bound for $d$ such that the higher-degree Manin-Mumford property holds.

math.AG↗

arXiv2022

Derangements and the $p$-adic incomplete gamma function

We introduce a $p$-adic analogue of the incomplete gamma function. We also introduce quantities ($m$-values) associated to a function on natural numbers and prove a new characterization of $p$-adic continuity for functions with $p$-integral $m$-values. Combinatorial interpretations for the integral values of the incomplete gamma function and functions with $m$-values zero or one are obtained, which show that these functions count derangements in generalized symmetric groups and permutations with restricted cycle lengths.

math.NT↗