arXiv subjects

Kevin O'Bryant

Publications and source records attributed to Kevin O'Bryant.

At least 19 recordsLinked to original sources

arXiv2026

On the Thickness of Infinite Generalized Sidon Sets, I

Let $g \ge1$. A set $\mathcal{A}$ of nonnegative integers is a Sidon set if for each $d>0$ there is at most one pair $(a,b) \in \mathcal{A} \times \mathcal{A}$ with $d=a-b$. If there are at most $g$ pairs, then $\mathcal{A}$ is a $g$-Golomb ruler. We prove that if $\mathcal{A}$ is a $g$-Golomb ruler, then \[\liminf_{n\to\infty} \frac{ | \mathcal{A}\cap[0,n) | }{\sqrt{n/\log n}} \le \frac{2\sqrt g }{\sqrt{\log 2}},\] generalizing and sharpening results of Erdős and Cilleruelo. There is a $g$-Golomb ruler $\mathcal{G}$ with \[\frac{\sqrt g }{\sqrt2} \le \limsup_{n\to\infty} \frac{ | \mathcal{G}\cap[0,n) | }{\sqrt n} \le \sqrt{g } ,\] generalizing a result of Krückeberg.

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arXiv2026

On the Thickness of Infinite Generalized Sidon Sets, II

A set $\mathcal{A}$ of nonnegative integers is a $B_h$-set if the sums $a_1+\cdots+a_h$ with $a_1\le\cdots\le a_h$ and $a_i\in\mathcal{A}$ are distinct; a $B_2$-set is a Sidon set. We prove that for every even $h$ and every $B_h$-set $\mathcal{A}$, \[ \liminf_{n\to\infty} \frac{ | \mathcal{A}\cap[0,n) | }{\sqrt[h]{n/\log n}} \le \left(\fracπ{\log 2} \cdot \frac{Γ(1+h/2)^2}{Γ(1+1/h)^{h}}\right)^{1/h}. \]

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arXiv2026

Optimal Diameters of High Multiplicity g-Golomb Rulers

A set $\mathcal{G}$ of integers is called a $g$-Golomb ruler of length $n$ if the difference between any two distinct elements of $\mathcal{G}$ is repeated at most $g$ times. If $g=1$, these are also called $B_2$-sets, Sidon sets, and Babcock sets. We define $G(g,n)$ to represent the minimum diameter of a $g$-Golomb Ruler. In this paper, we prove that for all $b\ge 1$, if $g \ge \frac{7}{4}\left(b^{3/2} -b\right)+1,$ then $G(g,g+b)=g+2b-2$. Sharper bounds are given for $b\le 18$. The main technique is through an arithmetic property of the integers that are \emph{not} in a $g$-Golomb ruler, leading us to introduce LM rulers, a new class of rulers where every distance $d$ occurs as a difference at most $d-1$ times. We show that the minimum diameter of an $n$-element LM ruler $L(n)$ is $\sqrt{8/9} \cdot (n-1)^{3/2} \le L(n) \le \frac{7}{4}\left((n+1)^{3/2}-(n+1)\right).$

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arXiv2025

On Nathanson's Triangular Number Phenomenon

For a finite set $A\subseteq \mathbb{Z}$, the $h$-fold sumset is $hA :=\{x_1+\dots+x_h:x_i\in A\}$. We interpret the beginning of the sequence of sumset sizes $(|hA|)_{h=1}^\infty$ in terms of the successive $L^1$-minima of a lattice (specifically, the points in $\mathbb{Z}^{|A|}$ whose coordinates sum to 0 and which are perpendicular to $\langle a_1,\dots,a_{|A|}\rangle$). In particular, if $h_1,h_2$ are the first and second minima, and $1\le h<h_1$, then $|hA|=\binom{h+|A|-1}{|A|-1}$, while if $h_1\le h <h_2$, then $|hA|=\binom{h+|A|-1}{|A|-1}-\binom{h-h_1+|A|-1}{|A|-1}$. This explains the appearance of triangular numbers in the sequence of sumset sizes, an observation related to a recent experiment of Nathanson.

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arXiv2025

Visualizing the Sum-Product Conjecture

Let $SPP(n)$ be the set $\left\{\big(|A+A|,|A A|\big) : A\subseteq {\mathbb N}, |A|=n\right\}$ of sum-product pairs, where $A+A$ is the sumset $\{a+b : a,b\in A\}$ and $A A$ is the product set $\{ab:a,b\in A\}$. We construct a dataset consisting of 1162868 sets whose sum-product pairs are at least $84\%$ of $SPP(n)$ for each $n\le 32$. Notably, we do **not** see evidence in favor of Erdős's Sum-Product Conjecture in our dataset. For $n\le 6$, we prove the exact value of $SPP(n)$. We include a number of conjectures, open problems, and observations motivated by this dataset, a large number of color visualizations.

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arXiv2024

Bounds for Greedy $B_h$-sets

A set $A$ of nonnegative integers is called a $B_h$-set if every solution to $a_1+\dots+a_h = b_1+\dots+b_h$, where $a_i,b_i \in A$, has $\{a_1,\dots,a_h\}=\{b_1,\dots,b_h\}$ (as multisets). Let $γ_k(h)$ be the $k$-th positive element of the greedy $B_h$-set. We give a nontrivial lower bound on $γ_5(h)$, and a nontrivial upper bound on $γ_k(h)$ for $k\ge 5$. Specifically, $\frac 18 h^4 +\frac12 h^3 \le γ_5(h) \le 0.467214 h^4+O(h^3)$, although we conjecture that $γ_5(h)=\frac13 h^4 +O(h^3)$. We show that $γ_k(h) \ge \frac{1}{k!} h^{k-1} + O(h^{k-2})$ for $k\ge 1$ and $γ_k(h) \le α_k h^{k-1}+O(h^{k-2})$, where $α_6 := 0.382978$, $α_7 := 0.269877$, and for $k\ge 7$, $α_{k+1} := \frac{1}{2^k k!} \sum_{j=0}^{k-1} \binom{k-1}j\binom kj 2^j$. This work begins with a thorough introduction and concludes with a section of open problems.

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arXiv2024

Constructing Thick $B_h$-sets

A subset $A$ of a commutative semigroup $X$ is called a $B_h$ set in $X$ if the only solutions to $a_1+\dots+a_h = b_1 + \cdots +b_h$ (with $a_i,b_i \in A$) are the trivial solutions $\{a_1,\dots,a_h\} = \{b_1,\dots,b_h\}$ (as multisets). With $h=2$ and $X={\mathbb Z}$, these sets are also known as Sidon sets, Golomb Rulers, and Babcock sets. In this work, we generalize constructions of Bose-Chowla and Singer and give the resultant bounds on the diameter of a $k$ element $B_h$ set in $\mathbb Z$ for small $k$. We conclude with a list of open problems.

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arXiv2023

The fourth positive element in the greedy $B_h$-set

For $h \geq 1$, a $B_h$-set is a set of integers such that every integer $n$ has at most one representation in the form $n = a_{i_1} + \cdots + a_{i_h}$, where $a_{i_r} \in A$ for all $r = 1,\ldots, h$ and $a_{i_1} \leq \ldots \leq a_{i_h}$. The greedy $B_h$-set is the infinite set of nonnegative integers $\{a_0(h), a_1(h), a_2(h), \ldots \}$ constructed as follows: If $a_0(h) = 0$ and $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h) \}$ is a $B_h$-set, then $a_{k+1}(h)$ is the least positive integer such that $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h), a_{k+1}(h) \}$ is a $B_h$-set. Then $a_1(h) = 1$, $a_2(h) = h+1$, and $a_3(h) = h^2+h+1$ for all $h$. This paper proves that $a_4(h)$, the fourth term of the greedy $B_h$-set is $\left( h^3 + 3h^2 + 3h + 1\right) /2$ if $h$ is odd and $\left( h^3 + 2h^2 + 3h + 2\right) /2$ if $h$ is even.

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arXiv2023

On the Diameter of Finite Sidon Sets

We prove that the diameter of a Sidon set (also known as a Babcock sequence, Golomb ruler, or $B_2$ set) with $k$ elements is at least $k^2-b k^{3/2}-O(k)$ where $b\le 1.96365$, a comparatively large improvement on past results. Equivalently, a Sidon set with diameter $n$ has at most $n^{1/2}+0.98183n^{1/4}+O(1)$ elements. The proof is conceptually simple but very computationally intensive, and the proof uses substantial computer assistance. We also provide a proof of $b\le 1.99058$ that can be verified by hand, which still improves on past results. Finally, we prove that $g$-thin Sidon sets (aka $g$-Golomb rulers) with $k$ elements have diameter at least $g^{-1} k^2 - (2-\varepsilon)g^{-1}k^{3/2} - O(k)$, with $\varepsilon\ge 0.02g^{-2}$.

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arXiv2022

Counting Dope Matrices

For a polynomial $P$ of degree $n$ and an $m$-tuple $Λ=(λ_1,\dots,λ_m)$ of distinct complex numbers, the dope matrix of $P$ with respect to $Λ$ is $D_P(Λ)=(δ_{ij})_{i\in [1,m],j\in[0,n]}$, where $δ_{ij}=1$ if $P^{(j)}(λ_i)=0$, and $δ_{ij}=0$ otherwise. Our first result is a combinatorial characterization of the $2$-row dope matrices (for all pairs $Λ$); using this characterization, we solve the associated enumeration problem. We also give upper bounds on the number of $m\times(n+1)$ dope matrices, and we show that the number of $m \times (n+1)$ dope matrices for a fixed $m$-tuple $Λ$ is maximized when $Λ$ is generic. Finally, we resolve an ``extension'' problem of Nathanson and present several open problems.

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arXiv2022

On the size of finite Sidon sets

A Sidon set is a set of integers containing no nontrivial solutions to the equation $a+b=c+d$. We improve on the lower bound on the diameter of a Sidon set with $k$ elements: if $k$ is sufficiently large and ${\cal A}$ is a Sidon set with $k$ elements, then $diam({\cal A})\ge k^2-1.99405 k^{3/2}$. Alternatively, if $n$ is sufficiently large, then the largest subset of $\{1,2,\dots,n\}$ that is a Sidon set has cardinality at most $n^{1/2}+0.99703 n^{1/4}$. While these are only slight numerical improvements on Balogh-Füredi-Roy (arXiv:2103:15850v2), we use a method that is logically simpler.

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arXiv2021

Multidimensional Padé approximation of binomial functions: Equalities

Let $ω_0,\dots,ω_M$ be complex numbers. If $H_0,\dots,H_M$ are polynomials of degree at most $ρ_0,\dots,ρ_M$, and $G(z)=\sum_{m=0} ^M H_m(z) (1-z)^{ω_m}$ has a zero at $z=0$ of maximal order (for the given $ω_m,ρ_m$), we say that $H_0,\dots,H_M$ are a \emph{multidimensional Padé approximation of binomial functions}, and call $G$ the Padé remainder. We collect here with proof all of the known expressions for $G$ and $H_m$, including a new one: the Taylor series of $G$. We also give a new criterion for systems of Padé approximations of binomial functions to be perfect (a specific sort of independence used in applications).

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arXiv2020

Counting Zeros of Dirichlet $L$-Functions

We give explicit upper and lower bounds for $N(T,χ)$, the number of zeros of a Dirichlet $L$-function with character $χ$ and height at most $T$. Suppose that $χ$ has conductor $q>1$, and that $T\geq 5/7$. If $\ell=\log\frac{q(T+2)}{2π}> 1.567$, then \begin{equation*} \left| N(T,χ) - \left( \frac{T}π \log\frac{qT}{2πe} -\frac{χ(-1)}{4}\right) \right| \le 0.22737 \ell + 2 \log(1+\ell) - 0.5. \end{equation*} We give slightly stronger results for small $q$ and $T$. Along the way, we prove a new bound on $|L(s,χ)|$ for $σ<-1/2$.

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arXiv2020

Representing Ordinal Numbers with Arithmetically Interesting Sets of Real Numbers

For a real number $x$ and set of natural numbers $A$, define $x \ast A := \{ x a \bmod 1: a\in A\}\subseteq [0,1).$ We consider relationships between $x$, $A$, and the order-type of $x\ast A$. For example, for every irrational $x$ and order-type $α$, there is an $A$ with $x\ast A \simeq α$, but if $α$ is a well order, then $A$ must be a thin set. If, however, $A$ is restricted to be a subset of the powers of 2, then not every order type is possible, although arbitrarily large countable well orders arise.

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arXiv2018

Explicit bounds for primes in arithmetic progressions

We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $θ(x;q,a)$ denotes the sum of the logarithms of the primes $p \equiv a \pmod{q}$ with $p \leq x$, we show that $$ \bigg| θ(x; q, a) - \frac{x}{ϕ(q)} \bigg| < \frac1{160} \frac{x}{\log x}, $$ for all $x \ge 8 \cdot 10^9$ (with sharper constants obtained for individual such moduli $q$). We establish inequalities of the same shape for the other standard prime-counting functions $π(x;q,a)$ and $ψ(x;q,a)$, as well as inequalities for the $n$th prime congruent to $a\pmod q$ when $q\le1200$. For moduli $q>10^5$, we find even stronger explicit inequalities, but only for much larger values of $x$. Along the way, we also derive an improved explicit lower bound for $L(1,χ)$ for quadratic characters $χ$, and an improved explicit upper bound for exceptional zeros.

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arXiv2015

Sets of natural numbers with proscribed subsets

Fix $A$, a family of subsets of natural numbers, and let $G_A(n)$ be the maximum cardinality of a subset of $\{1,2,..., n\}$ that does not have any subset in $A$. We consider the general problem of giving upper bounds on $G_A(n)$ and give some new upper bounds on some families that are closed under dilation. Specific examples include sets that do not contain any geometric progression of length $k$ with integer ratio, sets that do not contain any geometric progression of length $k$ with rational ratio, and sets of integers that do not contain multiplicative squares, i.e., nontrivial sets of the form $\{a, ar, as, ars\}$.

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arXiv2015

The sequence of fractional parts of roots

We study the function M(t,n) = Floor[ 1 / {t^(1/n)} ], where t is a positive real number, Floor[.] and {.} are the floor and fractional part functions, respectively. In a recent article in the Monthly, Nathanson proved that if log(t) is rational, then for all but finitely many positive integers n one has M(t,n) = Floor[ n / log(t) - 1/2 ]. We extend this by showing that, without condition on t, all but a zero-density set of integers n satisfy M(t,n) = Floor[ n / log(t) - 1/2 ]. Using a metric result of Schmidt, we show that almost all t have asymptotically log(t) log(x)/12 exceptional n<x. Using continued fractions, we produce uncountably many t that have only finitely many exceptional n, and also give uncountably many explicit t that have infinitely many exceptional n.

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arXiv2014

A problem of Rankin on sets without geometric progressions

A geometric progression of length $k$ and integer ratio is a set of numbers of the form $\{a,ar,\dots,ar^{k-1}\}$ for some positive real number $a$ and integer $r\geq 2$. For each integer $k \geq 3$, a greedy algorithm is used to construct a strictly decreasing sequence $(a_i)_{i=1}^{\infty}$ of positive real numbers with $a_1 = 1$ such that the set \[ G^{(k)} = \bigcup_{i=1}^{\infty} \left(a_{2i} , a_{2i-1} \right] \] contains no geometric progression of length $k$ and integer ratio. Moreover, $G^{(k)}$ is a maximal subset of $(0,1]$ that contains no geometric progression of length $k$ and integer ratio. It is also proved that there is a strictly increasing sequence $(A_i)_{i=1}^{\infty}$ of positive integers with $A_1 = 1$ such that $a_i = 1/A_i$ for all $i = 1,2,3,\ldots$. The set $G^{(k)}$ gives a new lower bound for the maximum cardinality of a subset of the set of integers $\{1,2,\dots,n\}$ that contains no geometric progression of length $k$ and integer ratio.

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