Zeckendorf's Theorem states that every positive integer can be uniquely represented as a sum of non-adjacent Fibonacci numbers, indexed from 1, 2, 3, 5, . . .. This has been generalized by many authors, in particular to constant coefficient fixed depth linear recurrences with positive (or in some cases non-negative) coefficients. In this work we extend this result to a recurrence with non-constant coefficients, an+1 = nan + an−1. The decomposition law becomes every m has a unique decomposition as siai with si ≤ i, where if si = i then si−1 = 0. Similar to Zeckendorf's original proof, we use the greedy algorithm. We show that almost all the gaps between summands, as n approaches infinity, are of length zero, and give a heuristic that the distribution of the number of summands tends to a Gaussian.Furthermore, we build a game based upon this recurrence relation, generalizing a game on the Fibonacci numbers. Given a fixed integer n and an initial decomposition of n = na1, the players alternate by using moves related to the recurrence relation, and whoever moves last wins. We show that the game is finite and ends at the unique decomposition of n, and that either player can win in a two-player game. We find the strategy to attain the shortest game possible, and the length of this shortest game. Then we show that in this generalized game when there are more than three players, no player has the winning strategy. Lastly, we demonstrate how one player in the two-player game can force the game to progress to their advantage.
The Erdős distance problem concerns the least number of distinct distances that can be determined by N points in the plane. The integer lattice with N points is known as near-optimal, as it spans around O(N/ log(N )) distinct distances which is the lower bound for a set of N points (Erdős, 1946). The only previous non-asymptotic work relating to the Erdős distance problem that has been done was carried out for N ≤ 13. We take a new non-asymptotic approach to this problem, studying the distance distribution, or in other words, the plot of frequencies of each distance of the N × N integer lattice. In order to fully characterize this distribution and determine its most common and least common distances, we adapt previous number-theoretic results from Fermat and Erdős, in order to relate the frequency of a given distance on the lattice to the sum-of-squares formula, which determines the number of ways in which a positive integer may be written as the sum of two squares.In order to apply our work on the lattice to the distance problem, we study the distance distributions of all its possible subsets; although this is a restricted case, we find that the structure of the integer lattice allows for the existence of subsets which can be chosen so that their distance distributions have certain properties, such as emulating the distribution of randomly distributed sets of points for certain small subsets, or that of the larger lattice itself. We define an error which compares the distance distribution of a subset with that of the full lattice. The structure of the integer lattice allows us to take subsets with certain geometric properties in order to maximize error, by exploiting the potential for sub-structure in the integer lattice. We show these geometric constructions explicitly; further, we calculate explicit upper bounds for the error for when the number of points in the subset is 4, 5, 9 or N 2 /2 and prove a lower bound for more general numbers of points.
Katz and Sarnak conjectured a correspondence between the n-level density statistics of zeros from families of L-functions with eigenvalues from random matrix ensembles, and in many cases the sums of smooth test functions, whose Fourier transforms are finitely supported over scaled zeros in a family, converge to an integral of the test function against a density W n,G depending on the symmetry G of the family (unitary, symplectic or orthogonal). This integral bounds the average order of vanishing at the central point of the corresponding family of Lfunctions.We can obtain better estimates on this vanishing in two ways. The first is to do more number theory, and prove results for larger n and greater support; the second is to do functional analysis and obtain better test functions to minimize the resulting integrals. We pursue the latter here when n = 2, minimizing 1 Φ(0, 0) R 2 W 2,G (x, y)Φ(x, y)dxdy over test functions Φ : R 2 → [0, ∞) with compactly supported Fourier transform. We study a restricted version of this optimization problem, imposing that our test functions take the form φ(x)ψ(y) for some fixed admissible ψ(y) and supp φ ⊆ [−1, 1]. Extending results from the 1-level case, namely the functional analytic arguments of Iwaniec, Luo and Sarnak and the differential equations method introduced by Freeman and Miller, we explicitly solve for the optimal φ for appropriately chosen fixed test function ψ. We conclude by discussing further improvements on estimates by the method of iteration.Contents
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