Author

# J. L. Duke

Bio: J. L. Duke is an academic researcher. The author has contributed to research in topics: Mathematics & Similarity (geometry). The author has co-authored 2 publications.

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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.

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TL;DR: In this paper , Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf provide bounds on the minimum number of distinct angles in general position in three dimensions.

Abstract: In 1946, Erd\H{o}s posed the distinct distance problem, which seeks to find
the minimum number of distinct distances between pairs of points selected from
any configuration of $n$ points in the plane. The problem has since been
explored along with many variants, including ones that extend it into higher
dimensions. Less studied but no less intriguing is Erd\H{o}s' distinct angle
problem, which seeks to find point configurations in the plane that minimize
the number of distinct angles. In their recent paper "Distinct Angles in
General Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf
use a logarithmic spiral to establish an upper bound of $O(n^2)$ on the minimum
number of distinct angles in the plane in general position, which prohibits
three points on any line or four on any circle.
We consider the question of distinct angles in three dimensions and provide
bounds on the minimum number of distinct angles in general position in this
setting. We focus on pinned variants of the question, and we examine explicit
constructions of point configurations in $\mathbb{R}^3$ which use
self-similarity to minimize the number of distinct angles. Furthermore, we
study a variant of the distinct angles question regarding distinct angle chains
and provide bounds on the minimum number of distinct chains in $\mathbb{R}^2$
and $\mathbb{R}^3$.