arXiv subjects

Steven J. Miller

Publications and source records attributed to Steven J. Miller.

At least 127 records · Page 7Linked to original sources

arXiv2021

A Refined Conjecture for the Variance of Gaussian Primes Across Sectors

We derive a refined conjecture for the variance of Gaussian primes across sectors, with a power saving error term, by applying the L-functions Ratios Conjecture. We observe a bifurcation point in the main term, consistent with the Random Matrix Theory (RMT) heuristic previously proposed by Rudnick and Waxman. Our model also identifies a second bifurcation point, undetected by the RMT model, that emerges upon taking into account lower order terms. For sufficiently small sectors, we moreover prove an unconditional result that is consistent with our conjecture down to lower order terms.

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arXiv2021

Distribution of Eigenvalues of Matrix Ensembles arising from Wigner and Palindromic Toeplitz Blocks

Random Matrix Theory (RMT) has successfully modeled diverse systems, from energy levels of heavy nuclei to zeros of $L$-functions; this correspondence has allowed RMT to successfully predict many number theoretic behaviors. However there are some operations which to date have no RMT analogue. Our motivation is to find an RMT analogue of Rankin-Selberg convolution, which constructs a new $L$-functions from an input pair. We report one such attempt; while it does not appear to model convolution, it does create new ensembles with properties hybridizing those of its constituents. For definiteness we concentrate on the ensemble of palindromic real symmetric Toeplitz (PST) matrices and the ensemble of real symmetric matrices, whose limiting spectral measures are the Gaussian and semi-circular distributions, respectively; these were chosen as they are the two extreme cases in terms of moment calculations. For a PST matrix $A$ and a real symmetric matrix $B$, we construct an ensemble of random real symmetric block matrices whose first row is $\lbrace A, B \rbrace$ and whose second row is $\lbrace B, A \rbrace$. By Markov's Method of Moments and the use of free probability, we show this ensemble converges weakly and almost surely to a new, universal distribution with a hybrid of Gaussian and semi-circular behaviors. We extend this construction by considering an iterated concatenation of matrices from an arbitrary pair of random real symmetric sub-ensembles with different limiting spectral measures. We prove that finite iterations converge to new, universal distributions with hybrid behavior, and that infinite iterations converge to the limiting spectral measure of the dominant component matrix.

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arXiv2021

Lower-Order Biases Second Moments of Dirichlet Coefficients in Families of $L$-Functions

Let $\mathcal E: y^2 = x^3 + A(T)x + B(T)$ be a nontrivial one-parameter family of elliptic curves over $\mathbb{Q}(T)$, with $A(T), B(T) \in \mathbb Z(T)$, and consider the $k$\textsuperscript{th} moments $A_{k,\mathcal{E}}(p) := \sum_{t (p)} a_{\mathcal{E}_t}(p)^k$ of the Dirichlet coefficients $a_{\mathcal{E}_t}(p) := p + 1 - |\mathcal{E}_t (\mathbb{F}_p)|$. Rosen and Silverman proved a conjecture of Nagao relating the first moment $A_{1,\mathcal{E}}(p)$ to the rank of the family over $\mathbb{Q}(T)$, and Michel proved that if $j(T)$ is not constant then the second moment is equal to $A_{2,\mathcal{E}}(p) = p^2 + O(p^{3/2})$. Cohomological arguments show that the lower order terms are of sizes $p^{3/2}, p, p^{1/2}$, and $1$. In every case we are able to analyze in closed form, the largest lower order term in the second moment expansion that does not average to zero is on average negative, though numerics suggest this may fail for families of moderate rank. We prove this Bias Conjecture for several large classes of families, including families with rank, complex multiplication, and constant $j(T)$-invariant. We also study the analogous Bias Conjecture for families of Dirichlet characters, holomorphic forms on GL$(2)/\mathbb{Q}$, and their symmetric powers and Rankin-Selberg convolutions. We identify all lower order terms in large classes of families, shedding light on the arithmetic objects controlling these terms. The negative bias in these lower order terms has implications toward the excess rank conjecture and the behavior of zeros near the central point in families of $L$-functions.

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arXiv2021

Tinkering with Lattices: A New Take on the Erdős Distance Problem

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 \textit{near-optimal}, as it spans $Θ(N/\sqrt{\log(N)})$ distinct distances, the lower bound for a set of $N$ points (Erdős, 1946). The only previous non-asymptotic work related to the Erdős distance problem that has been done was for $N \leq 13$. We take a new non-asymptotic approach to this problem in a model case, studying the distance distribution, or in other words, the plot of frequencies of each distance of the $N\times N$ integer lattice. In order to fully characterize this distribution, 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. We study the distance distributions of all the lattice's possible subsets; although this is a restricted case, 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 emulating 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; we show these geometric constructions explicitly. Further, we calculate explicit upper bounds for the error when the number of points in the subset is $4$, $5$, $9$ or $\left \lceil N^2/2\right\rceil$ and prove a lower bound in cases with a small number of points.

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arXiv2021

The Explicit Sato-Tate Conjecture For Primes In Arithmetic Progressions

Let $τ(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ Δ(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}τ(n) q^n \,,q=e^{2πi z} \] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's conjecture asserts that $τ(n)\neq 0$ for all $n\geq 1$; since $τ(n)$ is multiplicative, it suffices to study primes $p$ for which $τ(p)$ might possibly be zero. Assuming standard conjectures for the twisted symmetric power $L$-functions associated to $τ$ (including GRH), we prove that if $x\geq 10^{50}$, then \[ \#\{x < p\leq 2x: τ(p) = 0\} \leq 1.22 \times 10^{-5} \frac{x^{3/4}}{\sqrt{\log x}},\] a substantial improvement on the implied constant in previous work. To achieve this, under the same hypotheses, we prove an explicit version of the Sato-Tate conjecture for primes in arithmetic progressions.

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arXiv2020

Higher Order Fibonacci Sequences from Generalized Schreier sets

A Schreier set $S$ is a subset of the natural numbers with $\min S\ge |S|$. It has been known that the sequence $(a_{1,n})$, where $$a_{1,n}\ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n\mbox{ and } \min S \ge |S|\}|,$$ is the Fibonacci sequence. Generalizing this result, we prove that for all $p\in \mathbb{N}$, the sequence $(a_{p,n})$, where $$a_{p, n} \ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n\mbox{ and } \min S\ge p|S|\}|,$$ has a linear recurrence relation of higher order. We investigate further by requiring that ${\rm min}_2 S\ge q |S|$, where $\min_2 S$ is the second smallest element of $S$. We prove a linear recurrence relation for the sequence $(a_{p, q, n})$, where $$a_{p, q, n} \ :=\ |\{S\subseteq \mathbb{N}\,:\,\max S = n, \min S \ge p|S|\mbox{ and } {\rm min}_2 S\ge q|S|\}|,$$ and discuss a curious relationship between $(a_{q, n})$ and $(a_{p, q, n})$.

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arXiv2020

Generalizing Ruth-Aaron Numbers

Let $f(n)$ be the sum of the prime divisors of $n$, counted with multiplicity; thus $f(2020)$ $= f(2^2 \cdot 5 \cdot 101) = 110$. Ruth-Aaron numbers, or integers $n$ with $f(n)=f(n+1)$, have been an interest of many number theorists since the famous 1974 baseball game gave them the elegant name after two baseball stars. Many of their properties were first discussed by Erdös and Pomerance in 1978. In this paper, we generalize their results in two directions: by raising prime factors to a power and allowing a small difference between $f(n)$ and $f(n+1)$. We prove that the number of integers up to $x$ with $f_r(n)=f_r(n+1)$ is $O\left(\frac{x(\log\log x)^3\log\log\log x}{(\log x)^2}\right)$, where $f_r(n)$ is the Ruth-Aaron function replacing each prime factor with its $r-$th power. We also prove that the density of $n$ remains $0$ if $|f_r(n)-f_r(n+1)|\leq k(x)$, where $k(x)$ is a function of $x$ with relatively low rate of growth. Moreover, we further the discussion of the infinitude of Ruth-Aaron numbers and provide a few possible directions for future study.

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arXiv2020

Zeckendorf's Theorem Using Indices in an Arithmetic Progression

Zeckendorf's Theorem states that any positive integer can be uniquely decomposed into a sum of distinct, non-adjacent Fibonacci numbers. There are many generalizations, including results on existence of decompositions using only even indexed Fibonacci numbers. We extend these further and prove that similar results hold when only using indices in a given arithmetic progression. As part of our proofs, we generate a range of new recurrences for the Fibonacci numbers that are of interest in their own right.

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arXiv2020

Winning Strategy for the Multiplayer and Multialliance Zeckendorf Games

Edouard Zeckendorf proved that every positive integer $n$ can be uniquely written \cite{Ze} as the sum of non-adjacent Fibonacci numbers, known as the Zeckendorf decomposition. Based on Zeckendorf's decomposition, we have the Zeckendorf game for multiple players. We show that when the Zeckendorf game has at least $3$ players, none of the players have a winning strategy for $n\geq 5$. Then we extend the multi-player game to the multi-alliance game, finding some interesting situations in which no alliance has a winning strategy. This includes the two-alliance game, and some cases in which one alliance always has a winning strategy. %We examine what alliances, or combinations of players, can win, and what size they have to be in order to do so. We also find necessary structural constraints on what alliances our method of proof can show to be winning. Furthermore, we find some alliance structures which must have winning strategies. %We also extend the Generalized Zeckendorf game from $2$-players to multiple players. We find that when the game has $3$ players, player $2$ never has a winning strategy for any significantly large $n$. We also find that when the game has at least $4$ players, no player has a winning strategy for any significantly large $n$.

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arXiv2020

An Introduction to Completeness of Positive Linear Recurrence Sequences

A positive linear recurrence sequence (PLRS) is a sequence defined by a homogeneous linear recurrence relation with positive coefficients and a particular set of initial conditions. A sequence of positive integers is \emph{complete} if every positive integer is a sum of distinct terms of the sequence. One consequence of Zeckendorf's theorem is that the sequence of Fibonacci numbers is complete. Previous work has established a generalized Zeckendorf's theorem for all PLRS's. We consider PLRS's and want to classify them as complete or not. We study how completeness is affected by modifying the recurrence coefficients of a PLRS. Then, we determine in many cases which sequences generated by coefficients of the forms $[1, \ldots, 1, 0, \ldots, 0, N]$ are complete. Further, we conjecture bounds for other maximal last coefficients in complete sequences in other families of PLRS's. Our primary method is applying Brown's criterion, which says that an increasing sequence $\{H_n\}_{n = 1}^{\infty}$ is complete if and only if $H_1 = 1$ and $H_{n + 1} \leq 1 + \sum_{i = 1}^n H_i$. This paper is an introduction to the topic that is explored further in Completeness of Positive Linear Recurrence Sequences arXiv:2010.01655.

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arXiv2020

Extending Zeckendorf's Theorem to a Non-constant Recurrence and the Zeckendorf Game on this Non-constant Recurrence Relation

Zeckendorf's Theorem states that every positive integer can be uniquely represented as a sum of non-adjacent Fibonacci numbers, indexed from $1, 2, 3, 5,\ldots$. 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, $a_{n+1} = n a_{n} + a_{n-1}$. The decomposition law becomes every $m$ has a unique decomposition as $\sum s_i a_i$ with $s_i \le i$, where if $s_i = i$ then $s_{i-1} = 0$. 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 $n= na_1$, 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.

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arXiv2020

The limiting spectral measure for an ensemble of generalized checkerboard matrices

Random matrix theory successfully models many systems, from the energy levels of heavy nuclei to zeros of $L$-functions. While most ensembles studied have continuous spectral distribution, Burkhardt et al introduced the ensemble of $k$-checkerboard matrices, a variation of Wigner matrices with entries in generalized checkerboard patterns fixed and constant. In this family, $N-k$ of the eigenvalues are of size $O(\sqrt{N})$ and were called bulk while the rest are tightly contrained around a multiple of $N$ and were called blip. We extend their work by allowing the fixed entries to take different constant values. We can construct ensembles with blip eigenvalues at any multiples of $N$ we want with any multiplicity (thus we can have the blips occur at sequences such as the primes or the Fibonaccis). The presence of multiple blips creates technical challenges to separate them and to look at only one blip at a time. We overcome this by choosing a suitable weight function which allows us to localize at each blip, and then exploiting cancellation to deal with the resulting combinatorics to determine the average moments of the ensemble; we then apply standard methods from probability to prove that almost surely the limiting distributions of the matrices converge to the average behavior as the matrix size tends to infinity. For blips with just one eigenvalue in the limit we have convergence to a Dirac delta spike, while if there are $k$ eigenvalues in a blip we again obtain hollow $k \times k$ GOE behavior.

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arXiv2020

Bounds on Zeckendorf Games

Zeckendorf proved that every positive integer $n$ can be written uniquely as the sum of non-adjacent Fibonacci numbers. We use this decomposition to construct a two-player game. Given a fixed integer $n$ and an initial decomposition of $n=n F_1$, the two players alternate by using moves related to the recurrence relation $F_{n+1}=F_n+F_{n-1}$, and whoever moves last wins. The game always terminates in the Zeckendorf decomposition; depending on the choice of moves the length of the game and the winner can vary, though for $n\ge 2$ there is a non-constructive proof that Player 2 has a winning strategy. Initially the lower bound of the length of a game was order $n$ (and known to be sharp) while the upper bound was of size $n \log n$. Recent work decreased the upper bound to of size $n$, but with a larger constant than was conjectured. We improve the upper bound and obtain the sharp bound of $\frac{\sqrt{5}+3}{2}\ n - IZ(n) - \frac{1+\sqrt{5}}{2}Z(n)$, which is of order $n$ as $Z(n)$ is the number of terms in the Zeckendorf decomposition of $n$ and $IZ(n)$ is the sum of indices in the Zeckendorf decomposition of $n$ (which are at most of sizes $\log n$ and $\log^2 n$ respectively). We also introduce a greedy algorithm that realizes the upper bound, and show that the longest game on any $n$ is achieved by applying splitting moves whenever possible.

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arXiv2020

Constructions of Generalized MSTD Sets in Higher Dimensions

Let $A$ be a set of finite integers, define $$A+A \ = \ \{a_1+a_2: a_1,a_2 \in A\}, \ \ \ A-A \ = \ \{a_1-a_2: a_1,a_2 \in A\},$$ and for non-negative integers $s$ and $d$ define $$sA-dA\ =\ \underbrace{A+\cdots+A}_{s} -\underbrace{A-\cdots-A}_{d}.$$ A More Sums than Differences (MSTD) set is an $A$ where $|A+A| > |A-A|$. It was initially thought that the percentage of subsets of $[0,n]$ that are MSTD would go to zero as $n$ approaches infinity as addition is commutative and subtraction is not. However, in a surprising 2006 result, Martin and O'Bryant proved that a positive percentage of sets are MSTD, although this percentage is extremely small, about $10^{-4}$ percent. This result was extended by Iyer, Lazarev, Miller, ans Zhang [ILMZ] who showed that a positive percentage of sets are generalized MSTD sets, sets for $\{s_1,d_1\} \neq \{s_2, d_2\}$ and $s_1+d_1=s_2+d_2$ with $|s_1A-d_1A| > |s_2A-d_2A|$, and that in $d$-dimensions, a positive percentage of sets are MSTD. For many such results, establishing explicit MSTD sets in $1$-dimensions relies on the specific choice of the elements on the left and right fringes of the set to force certain differences to be missed while desired sums are attained. In higher dimensions, the geometry forces a more careful assessment of what elements have the same behavior as $1$-dimensional fringe elements. We study fringes in $d$-dimensions and use these to create new explicit constructions. We prove the existence of generalized MSTD sets in $d$-dimensions and the existence of $k$-generational sets, which are sets where $|cA+cA|>|cA-cA|$ for all $1\leq c \leq k$. We then prove that under certain conditions, there are no sets with $|kA+kA|>|kA-kA|$ for all $k \in \mathbb{N}.$

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arXiv2020

On the sum of $k$-th powers in terms of earlier sums

For $k$ a positive integer let $S_k(n) = 1^k + 2^k + \cdots + n^k$, i.e., $S_k(n)$ is the sum of the first $k$-th powers. Faulhaber conjectured (later proved by Jacobi) that for $k$ odd, $S_k(n)$ could be written as a polynomial of $S_1(n)$; for example $S_3(n) = S_1(n)^2$. We extend this result and prove that for any $k$ there is a polynomial $g_k(x,y)$ such that $S_k(n) = g(S_1(n), S_2(n))$. The proof yields a recursive formula to evaluate $S_k(n)$ as a polynomial of $n$ that has roughly half the number of terms as the classical one.

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arXiv2020

Deterministic Zeckendorf Games

Zeckendorf proved that every positive integer can be written uniquely as the sum of non-adjacent Fibonacci numbers. We further explore a two-player Zeckendorf game introduced in Baird-Smith, Epstein, Flint, and Miller: Given a fixed integer $n$ and an initial decomposition of $n = nF_1$, players alternate using moves related to the recurrence relation $F_{n+1} = F_n + F_{n_1}$, and the last player to move wins. We improve the upper bound on the number of moves possible and show that it is of the same order in $n$ as the lower bound; this is an improvement by a logarithm over previous work. The new upper bound is $3n - 3Z(n) - IZ(n) + 1$, and the existing lower bound is sharp at $n - Z(n)$ moves, where $Z(n)$ is the number of terms in the Zeckendorf decomposition of $n$ and $IZ(n)$ is the sum of indices in the same Zeckendorf decomposition of $n$. We also studied four deterministic variants of the game, where there was a fixed order on which available move one takes: Combine Largest, Split Largest, Combine Smallest and Split Smallest. We prove that Combine Largest and Split Largest realize the lower bound. Split Smallest has the largest number of moves over all possible games, and is close to the new upper bound. For Combine Split games, the number of moves grows linearly with $n$.

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arXiv2020

Central Limit Theorems for Compound Paths on the 2-Dimensional Lattice

Zeckendorf proved that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, and later researchers showed that the distribution of the number of summands needed for such decompositions of integers in $[F_n, F_{n+1})$ converges to a Gaussian as $n\to\infty$. Decomposition problems have been studied extensively for a variety of different sequences and notions of a legal decompositions; for the Fibonacci numbers, a legal decomposition is one for which each summand is used at most once and no two consecutive summands may be chosen. Recently, Chen et al. [CCGJMSY] generalized earlier work to $d$-dimensional lattices of positive integers; there, a legal decomposition is a path such that every point chosen had each component strictly less than the component of the previous chosen point in the path. They were able to prove Gaussianity results despite the lack of uniqueness of the decompositions; however, their results should hold in the more general case where some components are identical. The strictly decreasing assumption was needed in that work to obtain simple, closed form combinatorial expressions, which could then be well approximated and led to the limiting behavior. In this work we remove that assumption through inclusion-exclusion arguments. These lead to more involved combinatorial sums; using generating functions and recurrence relations we obtain tractable forms in $2$ dimensions and prove Gaussianity again; a more involved analysis should work in higher dimensions.

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arXiv2020

Distribution of missing differences in diffsets

Lazarev, Miller and O'Bryant investigated the distribution of $|S+S|$ for $S$ chosen uniformly at random from $\{0, 1, \dots, n-1\}$, and proved the existence of a divot at missing 7 sums (the probability of missing exactly 7 sums is less than missing 6 or missing 8 sums). We study related questions for $|S-S|$, and shows some divots from one end of the probability distribution, $P(|S-S|=k)$, as well as a peak at $k=4$ from the other end, $P(2n-1-|S-S|=k)$. A corollary of our results is an asymptotic bound for the number of complete rulers of length $n$.

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