arXiv subjects

Brian McDonald

Publications and source records attributed to Brian McDonald.

13 recordsLinked to original sources

arXiv2025

VC-dimension of Salem sets over finite fields

The VC-dimension, introduced by Vapnik and Chervonenkis in 1968 in the context of learning theory, has in recent years provided a rich source of problems in combinatorial geometry. Given $E\subseteq \mathbb{F}_q^d$ or $E\subseteq \mathbb{R}^d$, finding lower bounds on the VC-dimension of hypothesis classes defined by geometric objects such as spheres and hyperplanes is equivalent to constructing appropriate geometric configurations in $E$. The complexity of these configurations increases exponentially with the VC-dimension. These questions are related to the Erdős distance problem and the Falconer problem when considering a hypothesis class defined by spheres. In particular, the Erdős distance problem over finite fields is equivalent to showing that the VC-dimension of translates of a sphere of radius $t$ is at least one for all nonzero $t\in \mathbb{F}_q$. In this paper, we show that many of the existing techniques for distance problems over finite fields can be extended to a much broader context, not relying on the specific geometry of circles and spheres. We provide a unified framework which allows us to simultaneously study highly structured sets such as algebraic curves, as well as random sets.

math.CO↗

arXiv2025

The VC-dimension and point configurations in $\mathbb{R}^d$

Given a set $X$ and a collection ${\mathcal H}$ of functions from $X$ to $\{0,1\}$, the VC-dimension measures the complexity of the hypothesis class $\mathcal{H}$ in the context of PAC learning. In recent years, this has been connected to geometric configuration problems in vector spaces over finite fields. In particular, it is easy to show that the VC-dimension of the set of spheres of a given radius in $\mathbb{F}_q^d$ is equal to $d+1$, since this is how many points generically determine a sphere. It is known that for $E\subseteq \mathbb{F}_q^d$, $|E|\geq q^{d-\frac{1}{d-1}}$, the set of spheres centered at points in $E$, and intersected with the set $E$, has VC-dimension either $d$ or $d+1$. In this paper, we study a similar question over Euclidean space. We find an explicit dimensional threshold $s_d<d$ so that whenever $E\subseteq \mathbb{R}^d$, $d\geq 3$, and the Hausdorff dimension of $E$ is at least $s_d$, it follows that there exists an interval $I$ such that for any $t\in I$, the VC-dimension of the set of spheres of radius $t$ centered at points in $E$, and intersected with $E$, is at least $3$. In the process of proving this theorem, we also provide the first explicit dimensional threshold for a set $E\subseteq \mathbb{R}^3$ to contain a $4$-cycle, i.e. $x_1,x_2,x_3,x_4\in E$ satisfying $$ |x_1-x_2|=|x_2-x_3|=|x_3-x_4|=|x_4-x_1| $$

math.CA↗

arXiv2024

Interpolation of point configurations in the discrete plane

Defining distances over finite fields formally by $||x-y||:=(x_1-y_1)^2+\cdots + (x_d-y_d)^2$ for $x,y\in \mathbb{F}_q^d$, distance problems naturally arise in analogy to those studied by Erdős and Falconer in Euclidean space. Given a graph $G$ and a set $E\subseteq \mathbb{F}_q^2$, let $Δ_G(E)$ be the generalized distance set corresponding to $G$. In the case when $G$ is the complete graph on $k+1$ vertices, Bennett, Hart, Iosevich, Pakianathan, and Rudnev showed that when $|E|\geq q^{d-\frac{d-1}{k+1}}$, it follows that $|Δ_G(E)|\geq cq^{\binom{k+1}{2}}$. In the case when $k=d=2$, the threshold can be improved to $|E|\geq q^{\frac{8}{5}}$. Moreover, Jardine, Iosevich, and McDonald showed that in the case when $G$ is a tree with $k+1$ vertices, then whenever $E\subseteq \mathbb{F}_q^d$, $d\geq 2$ satisfies $|E|\geq C_kq^{\frac{d+1}{2}}$, it follows that $Δ_G(E)=\mathbb{F}_q^k$. In this paper, we present a technique which enables us to study certain graphs with both rigid and non-rigid components. In particular, we show that for $E\subseteq \mathbb{F}_q^2$, $q=p^n$, $n$ odd, $p\equiv 3 \ \text{mod} \ 4$, and $G$ is the graph consisting of two triangles joined at a vertex, then whenever $|E|\geq q^{\frac{12}{7}}$, it follows that $|Δ_G(E)|\geq cq^6$.

math.CO↗

arXiv2024

The VC-dimension of quadratic residues in finite fields

We study the Vapnik-Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, $\mathbb F_q$, when considered as a subset of the additive group. We conjecture that as $q \to \infty$, the squares have the maximum possible VC-dimension, viz. $(1+o(1))\log_2 q$. We prove, using the Weil bound for multiplicative character sums, that the VC-dimension is $\geq (\frac{1}{2} + o(1))\log_2 q$. We also provide numerical evidence for our conjectures. The results generalize to multiplicative subgroups $Γ\subseteq \mathbb F_q^\times$ of bounded index.

math.CO↗

arXiv2023

VC-Dimension and Distance Chains in $\mathbb{F}_q^d$

Given a domain $X$ and a collection $\mathcal{H}$ of functions $h:X\to \{0,1\}$, the Vapnik-Chervonenkis (VC) dimension of $\mathcal{H}$ measures its complexity in an appropriate sense. In particular, the fundamental theorem of statistical learning says that a hypothesis class with finite VC-dimension is PAC learnable. Recent work by Fitzpatrick, Wyman, the fourth and seventh named authors studied the VC-dimension of a natural family of functions $\mathcal{H}_t^{'2}(E): \mathbb{F}_q^2\to \{0,1\}$, corresponding to indicator functions of circles centered at points in a subset $E\subseteq \mathbb{F}_q^2$. They showed that when $|E|$ is large enough, the VC-dimension of $\mathcal{H}_t^{'2}(E)$ is the same as in the case that $E = \mathbb F_q^2$. We study a related hypothesis class, $\mathcal{H}_t^d(E)$, corresponding to intersections of spheres in $\mathbb{F}_q^d$, and ask how large $E\subseteq \mathbb{F}_q^d$ needs to be to ensure the maximum possible VC-dimension. We resolve this problem in all dimensions, proving that whenever $|E|\geq C_dq^{d-1/(d-1)}$ for $d\geq 3$, the VC-dimension of $\mathcal{H}_t^d(E)$ is as large as possible. We get a slightly stronger result if $d=3$: this result holds as long as $|E|\geq C_3 q^{7/3}$. Furthermore, when $d=2$ the result holds when $|E|\geq C_2 q^{7/4}$.

math.CO↗

arXiv2023

VC-Dimension of Hyperplanes over Finite Fields

Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements. For a subset $E\subseteq \mathbb{F}_q^d$ and a fixed nonzero $t\in \mathbb{F}_q$, let $\mathcal{H}_t(E)=\{h_y: y\in E\}$, where $h_y$ is the indicator function of the set $\{x\in E: x\cdot y=t\}$. Two of the authors, with Maxwell Sun, showed in the case $d=3$ that if $|E|\geq Cq^{\frac{11}{4}}$ and $q$ is sufficiently large, then the VC-dimension of $\mathcal{H}_t(E)$ is 3. In this paper, we generalize the result to arbitrary dimension and improve the exponent in the case $d=3$.

math.CO↗

arXiv2021

Cycles of arbitrary length in distance graphs on $\mathbb{F}_q^d$

For $E \subset {\Bbb F}_q^d$, $d \ge 2$, where ${\Bbb F}_q$ is the finite field with $q$ elements, we consider the distance graph ${\mathcal G}^{dist}_t(E)$, $t \not=0$, where the vertices are the elements of $E$, and two vertices $x$, $y$ are connected by an edge if $||x-y|| \equiv {(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=t$. We prove that if $|E| \ge C_k q^{\frac{d+2}{2}}$, then ${\mathcal G}^{dist}_t(E)$ contains a statistically correct number of cycles of length $k$. We are also going to consider the dot-product graph ${\mathcal G}^{prod}_t(E)$, $t \not=0$, where the vertices are the elements of $E$, and two vertices $x$, $y$ are connected by an edge if $x \cdot y \equiv x_1y_1+\dots+x_dy_d=t$. We obtain similar results in this case using more sophisticated methods necessitated by the fact that the function $x \cdot y$ is not translation invariant. The exponent $\frac{d+2}{2}$ is improved for sufficiently long cycles.

math.CO↗

arXiv2020

Distinct Distances Between a Circle and a Generic Set

Let $S$ be a set of points in $\mathbb{R}^2$ contained in a circle and $P$ an unrestricted point set in $\mathbb{R}^2$. We prove the number of distinct distances between points in $S$ and points in $P$ is at least $\min(|S||P|^{1/4-\varepsilon},|S|^{2/3}|P|^{2/3},|S|^2,|P|^2)$. This builds on work of Pach and De Zeeuw, Bruner and Sharir, McLaughlin and Omar and Mathialagan on distances between pairs of sets.

math.MG↗

arXiv2015

Gaps between zeros of GL(2) $L$-functions

Let $L(s,f)$ be an $L$-function associated to a primitive (holomorphic or Maass) cusp form $f$ on GL(2) over $\mathbb{Q}$. Combining mean-value estimates of Montgomery and Vaughan with a method of Ramachandra, we prove a formula for the mixed second moments of derivatives of $L(1/2+it,f)$ and, via a method of Hall, use it to show that there are infinitely many gaps between consecutive zeros of $L(s,f)$ along the critical line that are at least $\sqrt 3 = 1.732...$ times the average spacing. Using general pair correlation results due to Murty and Perelli in conjunction with a technique of Montgomery, we also prove the existence of small gaps between zeros of any primitive $L$-function of the Selberg class. In particular, when $f$ is a primitive holomorphic cusp form on GL(2) over $\mathbb{Q}$, we prove that there are infinitely many gaps between consecutive zeros of $L(s,f)$ along the critical line that are at most $< 0.823$ times the average spacing.

math.NT↗

arXiv2015

Gaussian Distribution of the Number of Summands in Generalized Zeckendorf Decompositions in Small Intervals

Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initial terms $F_1 = 1, F_2 = 2$. Previous work proved that as $n \to \infty$ the distribution of the number of summands in the Zeckendorf decompositions of $m \in [F_n, F_{n+1})$, appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in $[F_n, F_{n+1})$ share the same potential summands and hold for more general positive linear recurrence sequences $\{G_n\}$. We generalize these results to subintervals of $[G_n, G_{n+1})$ as $n \to \infty$ for certain sequences. The analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence $α(n) \to \infty$. As $n \to \infty$, for almost all $m \in [G_n, G_{n+1})$ the distribution of the number of summands in the generalized Zeckendorf decompositions of integers in the subintervals $[m, m + G_{α(n)})$, appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to $1$, $m$ has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and $[0, G_{α(n)})$ to obtain the result, since the summands are known to have Gaussian behavior in the latter interval.

math.NT↗

arXiv2014

Benford Behavior of Generalized Zeckendorf Decompositions

We prove connections between Zeckendorf decompositions and Benford's law. Recall that if we define the Fibonacci numbers by $F_1 = 1, F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$, every positive integer can be written uniquely as a sum of non-adjacent elements of this sequence; this is called the Zeckendorf decomposition, and similar unique decompositions exist for sequences arising from recurrence relations of the form $G_{n+1}=c_1G_n+\cdots+c_LG_{n+1-L}$ with $c_i$ positive and some other restrictions. Additionally, a set $S \subset \mathbb{Z}$ is said to satisfy Benford's law base 10 if the density of the elements in $S$ with leading digit $d$ is $\log_{10}{(1+\frac{1}{d})}$; in other words, smaller leading digits are more likely to occur. We prove that as $n\to\infty$ for a randomly selected integer $m$ in $[0, G_{n+1})$ the distribution of the leading digits of the summands in its generalized Zeckendorf decomposition converges to Benford's law almost surely. Our results hold more generally: one obtains similar theorems to those regarding the distribution of leading digits when considering how often values in sets with density are attained in the summands in the decompositions.

math.NT↗

arXiv2014

Benford Behavior of Zeckendorf Decompositions

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density of the elements in $S$ with leading digit $d$ is $\log_{10}{(1+\frac{1}{d})}$; in other words, smaller leading digits are more likely to occur. We prove that, as $n\to\infty$, for a randomly selected integer $m$ in $[0, F_{n+1})$ the distribution of the leading digits of the Fibonacci summands in its Zeckendorf decomposition converge to Benford's law almost surely. Our results hold more generally, and instead of looking at the distribution of leading digits one obtains similar theorems concerning how often values in sets with density are attained.

math.NT↗

arXiv2014

Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals

Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initial terms $F_1 = 1, F_2 = 2$. We consider the distribution of the number of summands involved in such decompositions. Previous work proved that as $n \to \infty$ the distribution of the number of summands in the Zeckendorf decompositions of $m \in [F_n, F_{n+1})$, appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in $[F_n, F_{n+1})$ share the same potential summands. We generalize these results to subintervals of $[F_n, F_{n+1})$ as $n \to \infty$; the analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence $α(n) \to \infty$. As $n \to \infty$, for almost all $m \in [F_n, F_{n+1})$ the distribution of the number of summands in the Zeckendorf decompositions of integers in the subintervals $[m, m + F_{α(n)})$, appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to $1$, $m$ has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and $[0, F_{α(n)})$ to obtain the result, since the summands are known to have Gaussian behavior in the latter interval. % We also prove the same result for more general linear recurrences.

math.NT↗