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Author

Justin Cheigh

Bio: Justin Cheigh is an academic researcher. The author has contributed to research in topics: Mathematics. The author has co-authored 4 publications.
Topics: Mathematics

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03 Oct 2022
TL;DR: For the generalized dihedral group D = Z 2 (cid:110) G, the authors showed that there are more MSTD sets than MDTS sets when 6 ≤ m ≤ c j √ n for c j = 1 . 3229 / √ 111 + 5 j , where j is the number of elements in G with order at most 2.
Abstract: . Given a group G , we say that a set A ⊆ G has more sums than differences (MSTD) if | A + A | > | A − A | , has more differences than sums (MDTS) if | A + A | < | A − A | , or is sum-difference balanced if | A + A | = | A − A | . A problem of recent interest has been to understand the frequencies of these type of subsets. The seventh author and Vissuet studied the problem for arbitrary finite groups G and proved that almost all subsets A ⊆ G are sum-difference balanced as | G | → ∞ . For the dihedral group D 2 n , they conjectured that of the remaining sets, most are MSTD, i.e., there are more MSTD sets than MDTS sets. Some progress on this conjecture was made by Haviland et al. in 2020, when they introduced the idea of partitioning the subsets by size: if, for each m , there are more MSTD subsets of D 2 n of size m than MDTS subsets of size m , then the conjecture follows. We extend the conjecture to generalized dihedral groups D = Z 2 (cid:110) G , where G is an abelian group of size n and the nonidentity element of Z 2 acts by inversion. We make further progress on the conjecture by considering subsets with a fixed number of rotations and reflections. By bounding the expected number of overlapping sums, we show that the collection S D,m of subsets of the generalized dihedral group D of size m has more MSTD sets than MDTS sets when 6 ≤ m ≤ c j √ n for c j = 1 . 3229 / √ 111 + 5 j , where j is the number of elements in G with order at most 2. We also analyze the expectation for | A + A | and | A − A | for A ⊆ D 2 n , proving an explicit formula for | A − A | when n is prime.
20 Oct 2022
TL;DR: In this paper , the authors studied the probabilistic aspects of random Zeckendorf games and showed that for any input N , the range of possible game lengths constitutes an interval of natural numbers: every game length between the shortest and longest game lengths can be achieved.
Abstract: . Zeckendorf proved that any positive integer has a unique decomposition as a sum of non-consecutive Fibonacci numbers, indexed by F 1 = 1 , F 2 = 2 , F n +1 = F n + F n − 1 . Motivated by this result, Baird, Epstein, Flint, and Miller [3] defined the two-player Zeckendorf game, where two players take turns acting on a multiset of Fibonacci numbers that always sums to N . The game terminates when no possible moves remain, and the final player to perform a move wins. Notably, [3] studied the setting of random games: the game proceeds by choosing an available move uniformly at random, and they conjecture that as the input N → ∞ , the distribution of random game lengths converges to a Gaussian. We prove that certain sums of move counts is constant, and find a lower bound on the number of shortest games on input N involving the Catalan numbers. The works [3] and Cuzensa et al. [5] determined how to achieve a shortest and longest possible Zeckendorf game on a given input N , respectively: we establish that for any input N , the range of possible game lengths constitutes an interval of natural numbers: every game length between the shortest and longest game lengths can be achieved. We further the study of probabilistic aspects of random Zeckendorf games. We study two probability measures on the space of all Zeckendorf games on input N : the uniform measure, and the measure induced by choosing moves uniformly at random at any given position. Under both measures that in the limit N → ∞ , both players win with probability 1 / 2. We also find natural partitions of the collection of all Zeckendorf games of a fixed input N , on which we observe weak convergence to a Gaussian in the limit N → ∞ . We conclude the work with many open problems.
06 Oct 2022
TL;DR: In particular, the authors showed that a hypothesis class with finite VC-dimension is PAC-learnable under the assumption that the VC dimension of the hypothesis class is as large as possible.
Abstract: 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 $|E|\geq C_2 q^{7/4}$.
19 Jul 2023
TL;DR: In this article , the authors generalize Sun's result to arbitrary dimension and improve the exponent in the case $d=3, where the VC-dimension of the vector space is 3.
Abstract: 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 $d=3$.