arXiv subjects

Steven J. Miller

Publications and source records attributed to Steven J. Miller.

At least 217 records · Page 12Linked to original sources

arXiv2014

The Weibull Distribution and Benford's Law

Benford's law states that many data sets have a bias towards lower leading digits (about $30\%$ are 1s). There are numerous applications, from designing efficient computers to detecting tax, voter and image fraud. It's important to know which common probability distributions are almost Benford. We show the Weibull distribution, for many values of its parameters, is close to Benford's law, quantifying the deviations. As the Weibull distribution arises in many problems, especially survival analysis, our results provide additional arguments for the prevalence of Benford behavior. The proof is by Poisson summation, a powerful technique to attack such problems.

math.PR↗

arXiv2014

Maass waveforms and low-lying zeros

The Katz-Sarnak Density Conjecture states that the behavior of zeros of a family of $L$-functions near the central point (as the conductors tend to zero) agrees with the behavior of eigenvalues near 1 of a classical compact group (as the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo and Sarnak proved that the behavior of zeros near the central point of holomorphic cusp forms agrees with the behavior of eigenvalues of orthogonal matrices for suitably restricted test functions $ϕ$. We prove similar results for families of cuspidal Maass forms, the other natural family of ${\rm GL}_2/\mathbb{Q}$ $L$-functions. For suitable weight functions on the space of Maass forms, the limiting behavior agrees with the expected orthogonal group. We prove this for $\Supp(\widehatϕ)\subseteq (-3/2, 3/2)$ when the level $N$ tends to infinity through the square-free numbers; if the level is fixed the support decreases to being contained in $(-1,1)$, though we still uniquely specify the symmetry type by computing the 2-level density.

math.NT↗

arXiv2014

Low-lying zeroes of Maass form $L$-functions

The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeros of families of automorphic $L$-functions agree with the scaling limits of eigenvalue distributions of classical subgroups of the unitary groups $U(N)$. This conjecture is often tested by way of computing particular statistics, such as the one-level density, which evaluates a test function with compactly supported Fourier transform at normalized zeros near the central point. Iwaniec, Luo, and Sarnak studied the one-level densities of cuspidal newforms of weight $k$ and level $N$. They showed in the limit as $kN \to\infty$ that these families have one-level densities agreeing with orthogonal type for test functions with Fourier transform supported in $(-2,2)$. Exceeding $(-1,1)$ is important as the three orthogonal groups are indistinguishable for support up to $(-1,1)$ but are distinguishable for any larger support. We study the other family of ${\rm GL}_2$ automorphic forms over $\mathbb{Q}$: Maass forms. To facilitate the analysis, we use smooth weight functions in the Kuznetsov formula which, among other restrictions, vanish to order $M$ at the origin. For test functions with Fourier transform supported inside $\left(-2 + \frac{2}{2M+1}, 2 - \frac{2}{2M+1}\right)$, we unconditionally prove the one-level density of the low-lying zeros of level 1 Maass forms, as the eigenvalues tend to infinity, agrees only with that of the scaling limit of orthogonal matrices.

math.NT↗

arXiv2014

Most Subsets are Balanced in Finite Groups

The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach's conjecture and Fermat's Last theorem) can be formulated in terms of the sumset $S + S = \{x+y : x,y\in S\}$ of a set of integers $S$. A finite set of integers $A$ is sum-dominated if $|A+A| > |A-A|$. Though it was believed that the percentage of subsets of $\{0,...,n\}$ that are sum-dominated tends to zero, in 2006 Martin and O'Bryant proved a very small positive percentage are sum-dominated if the sets are chosen uniformly at random (through work of Zhao we know this percentage is approximately $4.5 \cdot 10^{-4}$). While most sets are difference-dominated in the integer case, this is not the case when we take subsets of many finite groups. We show that if we take subsets of larger and larger finite groups uniformly at random, then not only does the probability of a set being sum-dominated tend to zero but the probability that $|A+A|=|A-A|$ tends to one, and hence a typical set is balanced in this case. The cause of this marked difference in behavior is that subsets of $\{0,..., n\}$ have a fringe, whereas finite groups do not. We end with a detailed analysis of dihedral groups, where the results are in striking contrast to what occurs for subsets of integers.

math.NT↗

arXiv2014

Sums and differences of correlated random sets

Many fundamental questions in additive number theory (such as Goldbach's conjecture, Fermat's last theorem, and the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair of elements contributes one sum and two differences, we expect that $|A-A| > |A+A|$ for a finite set $A$. However, in 2006 Martin and O'Bryant showed that a positive proportion of subsets of $\{0, \dots, n\}$ are sum-dominant, and Zhao later showed that this proportion converges to a positive limit as $n \to \infty$. Related problems, such as constructing explicit families of sum-dominant sets, computing the value of the limiting proportion, and investigating the behavior as the probability of including a given element in $A$ to go to zero, have been analyzed extensively. We consider many of these problems in a more general setting. Instead of just one set $A$, we study sums and differences of pairs of \emph{correlated} sets $(A,B)$. Specifically, we place each element $a \in \{0,\dots, n\}$ in $A$ with probability $p$, while $a$ goes in $B$ with probability $ρ_1$ if $a \in A$ and probability $ρ_2$ if $a \not \in A$. If $|A+B| > |(A-B) \cup (B-A)|$, we call the pair $(A,B)$ a \emph{sum-dominant $(p,ρ_1, ρ_2)$-pair}. We prove that for any fixed $\vecρ=(p, ρ_1, ρ_2)$ in $(0,1)^3$, $(A,B)$ is a sum-dominant $(p,ρ_1, ρ_2)$-pair with positive probability, and show that this probability approaches a limit $P(\vecρ)$. Furthermore, we show that the limit function $P(\vecρ)$ is continuous. We also investigate what happens as $p$ decays with $n$, generalizing results of Hegarty-Miller on phase transitions. Finally, we find the smallest sizes of MSTD pairs.

math.NT↗

arXiv2013

The Pythagorean Won-Loss Formula and Hockey: A Statistical Justification for Using the Classic Baseball Formula as an Evaluative Tool in Hockey

Originally devised for baseball, the Pythagorean Won-Loss formula estimates the percentage of games a team should have won at a particular point in a season. For decades, this formula had no mathematical justification. In 2006, Steven Miller provided a statistical derivation by making some heuristic assumptions about the distributions of runs scored and allowed by baseball teams. We make a similar set of assumptions about hockey teams and show that the formula is just as applicable to hockey as it is to baseball. We hope that this work spurs research in the use of the Pythagorean Won-Loss formula as an evaluative tool for sports outside baseball.

stat.AP↗

arXiv2013

Newman's conjecture in various settings

De Bruijn and Newman introduced a deformation of the Riemann zeta function $ζ(s)$, and found a real constant $Λ$ which encodes the movement of the zeros of $ζ(s)$ under the deformation. The Riemann hypothesis (RH) is equivalent to $Λ\le 0$. Newman made the conjecture that $Λ\ge 0$ along with the remark that "the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so." Newman's conjecture is still unsolved, and previous work could only handle the Riemann zeta function and quadratic Dirichlet $L$-functions, obtaining lower bounds very close to zero (for example, for $ζ(s)$ the bound is at least $-1.14541 \cdot 10^{-11}$, and for quadratic Dirichlet $L$-functions it is at least $-1.17 \cdot 10^{-7}$). We generalize the techniques to apply to automorphic $L$-functions as well as function field $L$-functions. We further determine the limit of these techniques by studying linear combinations of $L$-functions, proving that these methods are insufficient. We explicitly determine the Newman constants in various function field settings, which has strong implications for Newman's quantitative version of RH. In particular, let $\mathcal D \in \bbZ[T]$ be a square-free polynomial of degree 3. Let $D_p$ be the polynomial in $\bbF_p[T]$ obtained by reducing $\mathcal D$ modulo $p$. Then the Newman constant $Λ_{D_p}$ equals $\log \frac{|a_p(\mathcal D)|}{2\sqrt{p}}$; by Sato--Tate (if the curve is non-CM) there exists a sequence of primes such that $\lim_{n \to\infty} Λ_{D_{p_n}} = 0$. We end by discussing connections with random matrix theory.

math.NT↗

arXiv2013

Benford's Law and Continuous Dependent Random Variables

Many systems exhibit a digit bias. For example, the first digit base 10 of the Fibonacci numbers, or of $2^n$, equals 1 not 10% or 11% of the time, as one would expect if all digits were equally likely, but about 30% of the time. This phenomenon, known as Benford's Law, has many applications, ranging from detecting tax fraud for the IRS to analyzing round-off errors in computer science. The central question is determining which data sets follow Benford's law. Inspired by natural processes such as particle decay, our work examines models for the decomposition of conserved quantities. We prove that in many instances the distribution of lengths of the resulting pieces converges to Benford behavior as the number of divisions grow. The main difficulty is that the resulting random variables are dependent, which we handle by a careful analysis of the dependencies and tools from Fourier analysis to obtain quantified convergence rates.

math.PR↗

arXiv2013

Special Sets of Primes in Function Fields

When investigating the distribution of the Euler totient function, one encounters sets of primes P where if p is in P then r is in P for all r|(p-1). While it is easy to construct finite sets of such primes, the only infinite set known is the set of all primes. We translate this problem into the function field setting and construct an infinite such set in F_p[x] whenever p is equivalent to 2 or 5 modulo 9.

math.NT↗

arXiv2013

Generalizing Zeckendorf's Theorem to f-decompositions

A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, where $F_1 = 1$, $F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$. For general recurrences $\{G_n\}$ with non-negative coefficients, there is a notion of a legal decomposition which again leads to a unique representation, and the number of summands in the representations of uniformly randomly chosen $m \in [G_n, G_{n+1})$ converges to a normal distribution as $n \to \infty$. We consider the converse question: given a notion of legal decomposition, is it possible to construct a sequence $\{a_n\}$ such that every positive integer can be decomposed as a sum of terms from the sequence? We encode a notion of legal decomposition as a function $f:\N_0\to\N_0$ and say that if $a_n$ is in an "$f$-decomposition", then the decomposition cannot contain the $f(n)$ terms immediately before $a_n$ in the sequence; special choices of $f$ yield many well known decompositions (including base-$b$, Zeckendorf and factorial). We prove that for any $f:\N_0\to\N_0$, there exists a sequence $\{a_n\}_{n=0}^\infty$ such that every positive integer has a unique $f$-decomposition using $\{a_n\}$. Further, if $f$ is periodic, then the unique increasing sequence $\{a_n\}$ that corresponds to $f$ satisfies a linear recurrence relation. Previous research only handled recurrence relations with no negative coefficients. We find a function $f$ that yields a sequence that cannot be described by such a recurrence relation. Finally, for a class of functions $f$, we prove that the number of summands in the $f$-decomposition of integers between two consecutive terms of the sequence converges to a normal distribution.

math.NT↗

arXiv2013

On the spectral distribution of large weighted random regular graphs

McKay proved that the limiting spectral measures of the ensembles of $d$-regular graphs with $N$ vertices converge to Kesten's measure as $N\to\infty$. In this paper we explore the case of weighted graphs. More precisely, given a large $d$-regular graph we assign random weights, drawn from some distribution $\mathcal{W}$, to its edges. We study the relationship between $\mathcal{W}$ and the associated limiting spectral distribution obtained by averaging over the weighted graphs. Among other results, we establish the existence of a unique `eigendistribution', i.e., a weight distribution $\mathcal{W}$ such that the associated limiting spectral distribution is a rescaling of $\mathcal{W}$. Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is $O(1/d^2)$). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.

math.PR↗

arXiv2013

Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets

A More Sums Than Differences (MSTD) set is a set of integers A contained in {0, ..., n-1} whose sumset A+A is larger than its difference set A-A. While it is known that as n tends to infinity a positive percentage of subsets of {0, ..., n-1} are MSTD sets, the methods to prove this are probabilistic and do not yield nice, explicit constructions. Recently Miller, Orosz and Scheinerman gave explicit constructions of a large family of MSTD sets; though their density is less than a positive percentage, their family's density among subsets of {0, ..., n-1} is at least C/n^4 for some C>0, significantly larger than the previous constructions, which were on the order of 1 / 2^{n/2}. We generalize their method and explicitly construct a large family of sets A with |A+A+A+A| > |(A+A)-(A+A)|. The additional sums and differences allow us greater freedom than in Miller, Orosz and Scheinerman, and we find that for any epsilon>0 the density of such sets is at least C / n^epsilon. In the course of constructing such sets we find that for any integer k there is an A such that |A+A+A+A| - |A+A-A-A| = k, and show that the minimum span of such a set is 30.

math.NT↗

arXiv2013

When Generalized Sumsets are Difference Dominated

We study the relationship between the number of minus signs in a generalized sumset, $A+...+A-...-A$, and its cardinality; without loss of generality we may assume there are at least as many positive signs as negative signs. As addition is commutative and subtraction is not, we expect that for most $A$ a combination with more minus signs has more elements than one with fewer; however, recently Iyer, Lazarev, Miller and Zhang proved that a positive percentage of the time the combination with fewer minus signs can have more elements. Their analysis involves choosing sets $A$ uniformly at random from ${0,...,N}$; this is equivalent to independently choosing each element of ${0,...,N}$ to be in $A$ with probability 1/2. We investigate what happens when instead each element is chosen with probability $p(N)$, with $\lim_{N\to\infty} p(N) =0$. We prove that the set with more minus signs is larger with probability 1 as $N\to\infty$ if $p(N)=cN^{-δ}$ for $δ\ge\frac{h-1}{h}$, where $h$ is the number of total summands in $A+...+A-...-A$, and explicitly quantify their relative sizes. The results generalize earlier work of Hegarty and Miller, and we see a phase transition in the behavior of the cardinalities when $δ= \frac{h-1}{h}$.

math.NT↗

arXiv2012

Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles

The limiting distribution of eigenvalues of N x N random matrices has many applications. One of the most studied ensembles are real symmetric matrices with independent entries iidrv; the limiting rescaled spectral measure (LRSM) $\widetildeμ$ is the semi-circle. Studies have determined the LRSMs for many structured ensembles, such as Toeplitz and circulant matrices. These have very different behavior; the LRSM for both have unbounded support. Given a structured ensemble such that (i) each random variable occurs o(N) times in each row and (ii) the LRSM exists, we introduce a parameter to continuously interpolate between these behaviors. We fix a p in [1/2, 1] and study the ensemble of signed structured matrices by multiplying the (i,j)-th and (j,i)-th entries of a matrix by a randomly chosen epsilon_ij in {1, -1}, with Prob(epsilon_ij = 1) = p (i.e., the Hadamard product). For p = 1/2 we prove that the limiting signed rescaled spectral measure is the semi-circle. For all other p, the limiting measure has bounded (resp., unbounded) support if $\widetildeμ$ has bounded (resp., unbounded) support, and converges to $\widetildeμ$ as p -> 1. Notably, these results hold for Toeplitz and circulant matrix ensembles. The proofs are by the Method of Moments. The analysis involves the pairings of 2k vertices on a circle. The contribution of each in the signed case is weighted by a factor depending on p and the number of vertices involved in at least one crossing. These numbers appear in combinatorics and knot theory. The number of configurations with no vertices involved in a crossing is well-studied, and are the Catalan numbers. We prove similar formulas for configurations with up to 10 vertices in at least one crossing. We derive a closed-form expression for the expected value and determine the asymptotics for the variance for the number of vertices in at least one crossing.

math.PR↗

arXiv2012

Distribution of Missing Sums in Sumsets

For any finite set of integers X, define its sumset X+X to be {x+y: x, y in X}. In a recent paper, Martin and O'Bryant investigated the distribution of |A+A| given the uniform distribution on subsets A of {0, 1, ..., n-1}. They also conjectured the existence of a limiting distribution for |A+A| and showed that the expectation of |A+A| is 2n - 11 + O((3/4)^{n/2}). Zhao proved that the limits m(k) := lim_{n --> oo} Prob(2n-1-|A+A|=k) exist, and that sum_{k >= 0} m(k)=1. We continue this program and give exponentially decaying upper and lower bounds on m(k), and sharp bounds on m(k) for small k. Surprisingly, the distribution is at least bimodal; sumsets have an unexpected bias against missing exactly 7 sums. The proof of the latter is by reduction to questions on the distribution of related random variables, with large scale numerical computations a key ingredient in the analysis. We also derive an explicit formula for the variance of |A+A| in terms of Fibonacci numbers, finding Var(|A+A|) is approximately 35.9658. New difficulties arise in the form of weak dependence between events of the form {x in A+A}, {y in A+A}. We surmount these obstructions by translating the problem to graph theory. This approach also yields good bounds on the probability for A+A missing a consecutive block of length k.

math.NT↗

arXiv2012

The Average Gap Distribution for Generalized Zeckendorf Decompositions

An interesting characterization of the Fibonacci numbers is that, if we write them as $F_1 = 1$, $F_2 = 2$, $F_3 = 3$, $F_4 = 5, ...$, then every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. This is now known as Zeckendorf's theorem [21], and similar decompositions exist for many other sequences ${G_{n+1} = c_1 G_{n} + ... + c_L G_{n+1-L}}$ arising from recurrence relations. Much more is known. Using continued fraction approaches, Lekkerkerker [15] proved the average number of summands needed for integers in $[G_n, G_{n+1})$ is on the order of $C_{\rm Lek} n$ for a non-zero constant; this was improved by others to show the number of summands has Gaussian fluctuations about this mean. Kolo$\breve{\rm g}$lu, Kopp, Miller and Wang [17, 18] recently recast the problem combinatorially, reproving and generalizing these results. We use this new perspective to investigate the distribution of gaps between summands. We explore the average behavior over all $m \in [G_n, G_{n+1})$ for special choices of the $c_i$'s. Specifically, we study the case where each $c_i \in {0,1}$ and there is a $g$ such that there are always exactly $g-1$ zeros between two non-zero $c_i$'s; note this includes the Fibonacci, Tribonacci and many other important special cases. We prove there are no gaps of length less than $g$, and the probability of a gap of length $j > g$ decays geometrically, with the decay ratio equal to the largest root of the recurrence relation. These methods are combinatorial and apply to related problems; we end with a discussion of similar results for far-difference (i.e., signed) decompositions.

math.NT↗

arXiv2012

Coordinate sum and difference sets of $d$-dimensional modular hyperbolas

Many problems in additive number theory, such as Fermat's last theorem and the twin prime conjecture, can be understood by examining sums or differences of a set with itself. A finite set $A \subset \mathbb{Z}$ is considered sum-dominant if $|A+A|>|A-A|$. If we consider all subsets of ${0, 1, ..., n-1}$, as $n\to\infty$ it is natural to expect that almost all subsets should be difference-dominant, as addition is commutative but subtraction is not; however, Martin and O'Bryant in 2007 proved that a positive percentage are sum-dominant as $n\to\infty$. This motivates the study of "coordinate sum dominance". Given $V \subset (\Z/n\Z)^2$, we call $S:={x+y: (x,y) \in V}$ a coordinate sumset and $D:=\{x-y: (x,y) \in V\}$ a coordinate difference set, and we say $V$ is coordinate sum dominant if $|S|>|D|$. An arithmetically interesting choice of $V$ is $\bar{H}_2(a;n)$, which is the reduction modulo $n$ of the modular hyperbola $H_2(a;n) := {(x,y): xy \equiv a \bmod n, 1 \le x,y < n}$. In 2009, Eichhorn, Khan, Stein, and Yankov determined the sizes of $S$ and $D$ for $V=\bar{H}_2(1;n)$ and investigated conditions for coordinate sum dominance. We extend their results to reduced $d$-dimensional modular hyperbolas $\bar{H}_d(a;n)$ with $a$ coprime to $n$.

math.NT↗

arXiv2012

Quadratic fields with cyclic 2-class groups

For any integer $k\ge 1$, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order $2^k$. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class to be a square.

math.NT↗