publications | Caleb M. Shor, Ph.D.

2024

On the residues of rounded fractions with a common numerator

N. Dent, and C. Shor

Journal of Integer Sequences, 2024

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The proportion of integers \[ ⌊n/1⌋, ⌊n/2⌋, …, ⌊n/n⌋\]that are odd is asymptotically \( \log2 \). If instead of the floor function, one uses the nearest-integer function, the proportion drops to \( \pi/2-1 \). In this work, we prove these facts and, more generally, give an integral formula for the proportion of \[ ⌊(n-ν)/1+α⌋, ⌊(n-ν)/2+α⌋, …, ⌊(n-ν)/n+α⌋\]that are congruent to \(r \) modulo \(m \). We conclude by linking this problem to the Dirichlet divisor and Gauss circle problems, and counting lattice points in other quadratic regions.
@article{DentShor2024, author = {Dent, N. and Shor, C.}, title = {On the residues of rounded fractions with a common numerator}, journal = {Journal of Integer Sequences}, volume = {27}, year = {2024}, pages = {Article 24.2.5}, }

2023

Reflective numerical semigroups

Caleb M. Shor

Albanian J. Math., 2023

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We define a reflective numerical semigroup of genus \(g \) as a numerical semigroup that has a certain reflective symmetry when viewed within \( \mathbb Z \) as an array with \(g \) columns. Equivalently, a reflective numerical semigroup has one gap in each residue class modulo \(g \). In this paper, we give an explicit description for all reflective numerical semigroups. With this, we can describe the reflective members of well-known families of numerical semigroups as well as obtain formulas for the number of reflective numerical semigroups of a given genus or given Frobenius number.
@article{Shor23, author = {Shor, Caleb M.}, title = {Reflective numerical semigroups}, journal = {Albanian J. Math.}, fjournal = {Albanian Journal of Mathematics}, volume = {17}, year = {2023}, number = {1}, pages = {55--82}, issn = {1930-1235}, mrclass = {20M14 (11D07)}, mrnumber = {4544845}, doi = {10.51286/albjm/1675941730}, url = {https://doi.org/10.51286/albjm/1675941730}, }

2022

Equidistribution of numerical semigroup gaps modulo \(m \)

Caleb M. Shor

Discrete Mathematics, 2022

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For a positive integer \(m\), a finite set of integers is said to be equidistributed modulo \(m \) if the set contains an equal number of elements in each congruence class modulo \(m \). In this paper, we consider the problem of determining when the set of gaps of a numerical semigroup \(S \) is equidistributed modulo \(m \). Of particular interest is the case when the nonzero elements of an Apéry set of \(S \) form an arithmetic sequence. We explicitly describe such numerical semigroups \(S\) and determine conditions for which the sets of gaps of these numerical semigroups are equidistributed modulo \(m \).
@article{Shor22, author = {Shor, Caleb M.}, title = {Equidistribution of numerical semigroup gaps modulo \( m \)}, journal = {Discrete Mathematics}, volume = {345}, number = {10}, year = {2022}, url = {https://doi.org/10.1016/j.disc.2022.112995}, doi = {10.1016/j.disc.2022.112995}, }

2019

Characterizations of numerical semigroup complements via Apéry sets

T. Alden Gassert, and Caleb M. Shor

Semigroup Forum, 2019

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In this paper, we generalize the work of Tuenter to give an identity which completely characterizes the complement of a numerical semigroup in terms of its Apéry sets. Using this result, we compute the \(m\)th power Sylvester and alternating Sylvester sums for free numerical semigroups. Explicit formulas are given for small \(m\).
@article{GassertShor19, author = {Gassert, T. Alden and Shor, Caleb M.}, title = {Characterizations of numerical semigroup complements via {A}p{\'e}ry sets}, journal = {Semigroup Forum}, year = {2019}, volume = {98}, number = {1}, pages = {31--47}, issn = {1432-2137}, doi = {10.1007/s00233-018-9935-4}, url = {https://doi.org/10.1007/s00233-018-9935-4}, }
On free numerical semigroups and the construction of minimal telescopic sequences

Caleb M. Shor

Journal of Integer Sequences, 2019

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A free numerical semigroup is a submonoid of the non-negative integers with finite complement that is additively generated by the terms in a telescopic sequence with gcd 1. However, such a sequence need not be minimal, which is to say that some proper subsequence may generate the same numerical semigroup, and that subsequence need not be telescopic. In this paper, we will see that for a telescopic sequence with any gcd, there is a minimal telescopic sequence that generates the same submonoid. In particular, given a free numerical semigroup we can construct a telescopic generating sequence that is minimal. In the process, we will examine some operations on and constructions of telescopic sequences in general.
@article{Shor19, author = {Shor, Caleb M.}, title = {On free numerical semigroups and the construction of minimal telescopic sequences}, journal = {Journal of Integer Sequences}, volume = {22}, year = {2019}, pages = {Article 19.2.4}, }

2018

Higher-order Weierstrass weights of branch points on superelliptic curves

Caleb M. Shor

In Higher genus curves in mathematical physics and arithmetic geometry , 2018

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@incollection{Shor18,
  author = {Shor, Caleb M.},
  title = {Higher-order {W}eierstrass weights of branch points on superelliptic curves},
  booktitle = {Higher genus curves in mathematical physics and arithmetic geometry},
  publisher = {Amer. Math. Soc., Providence, RI},
  series = {Contemporary Mathematics},
  url = {https://arxiv.org/abs/1612.02730},
  doi = {10.1090/conm/703/14135},
  year = {2018},
  isbn = {978-1-4704-2856-3},
  volume = {703},
  pages = {143--156}
}

2017

2-Weierstrass points of genus 3 hyperelliptic curves with extra involutions

Tony Shaska, and Caleb M. Shor

Comm. Algebra, 2017

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On Sylvester sums of compound sequence semigroup complements

T. Alden Gassert, and Caleb M. Shor

J. Number Theory, 2017

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In this paper, we consider the set \(NR(G) \) of natural numbers which are not in the numerical semigroup generated by a compound sequence \(G\). We generalize a result of Tuenter which completely characterizes \(NR(G) \). We use this result to compute Sylvester sums, and we give a direct application to the computation of weights of higher-order Weierstrass points on some families of complex algebraic curves.
@article{GassertShor17, title = {On {S}ylvester sums of compound sequence semigroup complements}, journal = {J. Number Theory}, fjournal = {Journal of Number Theory}, volume = {180}, pages = {45--72}, year = {2017}, issn = {0022-314X}, doi = {10.1016/j.jnt.2017.03.025}, url = {http://dx.doi.org/10.1016/j.jnt.2017.03.025}, author = {Gassert, T. Alden and Shor, Caleb M.}, keywords = {Superelliptic curves} }

2015

Weierstrass points of superelliptic curves

T. Shaska, and C. Shor

In Advances on Superelliptic Curves and Their Applications , 2015

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Theta functions and symmetric weight enumerators for codes over imaginary quadratic fields

T. Shaska, and C. Shor

Des. Codes Cryptogr., 2015

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2014

On Jacobians of curves with superelliptic components

L. Beshaj, T. Shaska, and C. Shor

In Riemann and Klein surfaces, automorphisms, symmetries and moduli spaces , 2014

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2011

Genus calculations for towers of functions fields arising from equations of \( C_{ab} \) curves

Caleb M. Shor

Albanian J. Math., 2011

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2010

Codes over rings of size \(p^2 \) and lattices over imaginary quadratic fields

T. Shaska, C. Shor, and S. Wijesiri

Finite Fields Appl., 2010

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2007

Codes over \( \mathbb F_{p^2} \) and \( \mathbb F_p\times\mathbb F_p \), lattices, and theta functions

T. Shaska, and C. Shor

In Advances in coding theory and cryptography , 2007

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On the construction of codes from an asymptotically good tower over \( \mathbb F_8 \)

Caleb McKinley Shor

Serdica J. Comput., 2007

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2005

On towers of function fields and the construction of the corresponding Goppa codes

Caleb M. Shor

2005

Dissertation (Ph.D.)–Boston University

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