arXiv subjects

Steven J. Miller

Publications and source records attributed to Steven J. Miller.

At least 181 records · Page 10Linked to original sources

arXiv2016

On the Asymptotic Behavior of Variance of PLRS Decompositions

A positive linear recurrence sequence is of the form $H_{n+1} = c_1 H_n + \cdots + c_L H_{n+1-L}$ with each $c_i \ge 0$ and $c_1 c_L > 0$, with appropriately chosen initial conditions. There is a notion of a legal decomposition (roughly, given a sum of terms in the sequence we cannot use the recurrence relation to reduce it) such that every positive integer has a unique legal decomposition using terms in the sequence; this generalizes the Zeckendorf decomposition, which states any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. Previous work proved not only that a decomposition exists, but that the number of summands $K_n(m)$ in legal decompositions of $m \in [H_n, H_{n+1})$ converges to a Gaussian. Using partial fractions and generating functions it is easy to show the mean and variance grow linearly in $n$: $a n + b + o(1)$ and $C n + d + o(1)$, respectively; the difficulty is proving $a$ and $C$ are positive. Previous approaches relied on delicate analysis of polynomials related to the generating functions and characteristic polynomials, and is algebraically cumbersome. We introduce new, elementary techniques that bypass these issues. The key insight is to use induction and bootstrap bounds through conditional probability expansions to show the variance is unbounded, and hence $C > 0$ (the mean is handled easily through a simple counting argument).

math.NT↗

arXiv2016

New Behavior in Legal Decompositions Arising from Non-positive Linear Recurrences

Zeckendorf's theorem states every positive integer has a unique decomposition as a sum of non-adjacent Fibonacci numbers. This result has been generalized to many sequences $\{a_n\}$ arising from an integer positive linear recurrence, each of which has a corresponding notion of a legal decomposition. Previous work proved the number of summands in decompositions of $m \in [a_n, a_{n+1})$ becomes normally distributed as $n\to\infty$, and the individual gap measures associated to each $m$ converge to geometric random variables, when the leading coefficient in the recurrence is positive. We explore what happens when this assumption is removed in two special sequences. In one we regain all previous results, including unique decomposition; in the other the number of legal decompositions exponentially grows and the natural choice for the legal decomposition (the greedy algorithm) only works approximately 92.6\% of the time (though a slight modification always works). We find a connection between the two sequences, which explains why the distribution of the number of summands and gaps between summands behave the same in the two examples. In the course of our investigations we found a new perspective on dealing with roots of polynomials associated to the characteristic polynomials. This allows us to remove the need for the detailed technical analysis of their properties which greatly complicated the proofs of many earlier results in the subject, as well as handle new cases beyond the reach of existing techniques.

math.CO↗

arXiv2016

Legal Decompositions Arising from Non-positive Linear Recurrences

Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with leading term positive. We investigate legal decompositions arising from two new sequences: the $(s,b)$-Generacci sequence and the Fibonacci Quilt sequence. Both satisfy recurrence relations with leading term zero, and thus previous results and techniques do not apply. These sequences exhibit drastically different behavior. We show that the $(s,b)$-Generacci sequence leads to unique legal decompositions, whereas not only do we have non-unique legal decompositions with the Fibonacci Quilt sequence, we also have that in this case the average number of legal decompositions grows exponentially. Another interesting difference is that while in the $(s,b)$-Generacci case the greedy algorithm always leads to a legal decomposition, in the Fibonacci Quilt setting the greedy algorithm leads to a legal decomposition (approximately) 93\% of the time. In the $(s,b)$-Generacci case, we again have Gaussian behavior in the number of summands as well as for the Fibonacci Quilt sequence when we restrict to decompositions resulting from a modified greedy algorithm.

math.CO↗

arXiv2016

A Probabilistic Approach to Generalized Zeckendorf Decompositions

Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-$b$ expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts.

math.PR↗

arXiv2016

Some Results in the Theory of Low-lying Zeros: Determining the 1-level density, identifying the group symmetry and the arithmetic of moments of Satake parameters

While Random Matrix Theory has successfully modeled many quantities of families of L-functions, it frequently cannot see the family's arithmetic. In some situations this requires an extended theory that inserts arithmetic factors depending on the family, while in other cases these factors result in contributions which vanish in the limit, and are thus not detected. We review the general theory associated to one of the most important statistics, the n-level density of zeros near the central point. According to the Katz-Sarnak density conjecture, to each family of L-functions there is a corresponding symmetry group such that the behavior of zeros near the central point as the conductors tend to infinity agrees with the behavior of eigenvalues near 1 as the matrix size tends to infinity. We show how these calculations are done, emphasizing the techniques, methods and obstructions to improving the results, by considering in full detail a family of Dirichlet characters. We then describe how we may associate a symmetry constant to each family, and how to determine the symmetry group of a compound family in terms of the symmetries of the constituents. These calculations explain the remarkable universality of behavior, where the main terms are independent of the arithmetic (only the first two moments of the Satake parameters contribute in the limit; similar to the Central Limit Theorem, the higher moments are only felt in the rate of convergence). We end by exploring lower order terms in families of elliptic curves. We present evidence supporting a conjecture that the average second moment in one-parameter families without complex multiplication has, when appropriately viewed, a negative bias, and end with a discussion of the consequences of this bias on the distribution of low-lying zeros, in particular relations between such a bias and the observed excess rank in families.

math.NT↗

arXiv2016

Subsets of $\mathbb{F}_q[x]$ free of 3-term geometric progressions

Several recent papers have considered the Ramsey-theoretic problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to number fields, determining bounds on the greatest possible density of ideals avoiding geometric progressions. We study the analogous problem over $\mathbb{F}_q[x]$, first constructing a set greedily which avoids these progressions and calculating its density, and then considering bounds on the upper density of subsets of $\mathbb{F}_q[x]$ which avoid 3-term geometric progressions. This new setting gives us a parameter $q$ to vary and study how our bounds converge to 1 as it changes, and finite characteristic introduces some extra combinatorial structure that increases the tractibility of common questions in this area.

math.NT↗

arXiv2015

A Generalization of Zeckendorf's Theorem via Circumscribed $m$-gons

Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$, $F_1=1$ and $F_2=2$. The distribution of the number of summands in such decomposition converges to a Gaussian, the gaps between summands converges to geometric decay, and the distribution of the longest gap is similar to that of the longest run of heads in a biased coin; these results also hold more generally, though for technical reasons previous work needed to assume the coefficients in the recurrence relation are non-negative and the first term is positive. We extend these results by creating an infinite family of integer sequences called the $m$-gonal sequences arising from a geometric construction using circumscribed $m$-gons. They satisfy a recurrence where the first $m+1$ leading terms vanish, and thus cannot be handled by existing techniques. We provide a notion of a legal decomposition, and prove that the decompositions exist and are unique. We then examine the distribution of the number of summands used in the decompositions and prove that it displays Gaussian behavior. There is geometric decay in the distribution of gaps, both for gaps taken from all integers in an interval and almost surely in distribution for the individual gap measures associated to each integer in the interval. We end by proving that the distribution of the longest gap between summands is strongly concentrated about its mean, behaving similarly as in the longest run of heads in tosses of a coin.

math.NT↗

arXiv2015

Geometric-progression-free sets over quadratic number fields

A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets "greedily," a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.

math.NT↗

arXiv2015

Determining Optimal Test Functions for Bounding the Average Rank in Families of $L$-Functions

Given an $L$-function, one of the most important questions concerns its vanishing at the central point; for example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing there of an elliptic curve $L$-function equals the rank of the Mordell-Weil group. The Katz and Sarnak Density Conjecture states that this and other behavior is well-modeled by random matrix ensembles. This correspondence is known for many families when the test functions are suitably restricted. For appropriate choices, we obtain bounds on the average order of vanishing at the central point in families. In this note we report on progress in determining the optimal test functions for the various classical compact groups for different support restrictions, and discuss how this relates to improved rank bounds.

math.NT↗

arXiv2015

Crescent configurations

In 1989, Erdős conjectured that for a sufficiently large $n$ it is impossible to place $n$ points in general position in a plane such that for every $1\le i \le n-1$ there is a distance that occurs exactly $i$ times. For small $n$ this is possible and in his paper he provided constructions for $n\leq 8$. The one for $n=5$ was due to Pomerance while Palásti came up with the constructions for $n=7,8$. Constructions for $n=9$ and above remain undiscovered, and little headway has been made toward a proof that for sufficiently large $n$ no configuration exists. In this paper we consider a natural generalization to higher dimensions and provide a construction which shows that for any given $n$ there exists a sufficiently large dimension $d$ such that there is a configuration in $d$-dimensional space meeting Erdős' criteria.

math.CO↗

arXiv2015

The M&M Game: From Morsels to Modern Mathematics

To an adult, it's obvious that the day of someone's death is not precisely determined by the day of birth, but it's a very different story for a child. When the third named author was four years old he asked his father, the fifth named author: If two people are born on the same day, do they die on the same day? While this could easily be demonstrated through murder, such a proof would greatly diminish the possibility of teaching additional lessons, and thus a different approach was taken. With the help of the fourth named author they invented what we'll call \emph{the M\&M Game}: Given $k$ people, each simultaneously flips a fair coin, with each eating an M\&M on a head and not eating on a tail. The process then continues until all \mandms\ are consumed, and two people are deemed to die at the same time if they run out of \mandms\ together\footnote{Is one really living without \mandms?}. This led to a great concrete demonstration of randomness appropriate for little kids; it also led to a host of math problems which have been used in probability classes and math competitions. There are many ways to determine the probability of a tie, which allow us in this article to use this problem as a springboard to a lot of great mathematics, including memoryless process, combinatorics, statistical inference, graph theory, and hypergeometric functions.

math.HO↗

arXiv2015

Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions

Two well studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression, or in geometric progression. We study generalizations of this problem, by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence $x, f_n(x), f_n(f_n(x))$, where $f_n(x) = x + n$, $n$ a nonzero integer. Thus avoiding three-term arithmetic progressions is equivalent to containing no three elements of the form $x, f_n(x), f_n(f_n(x))$ with $f_n \in\mathcal{F}_{\rm t}$, the set of integer translations. One can similarly construct related progressions using different families of functions. We investigate several such families, including geometric progressions ($f_n(x) = nx$ with $n > 1$ a natural number) and exponential progressions ($f_n(x) = x^n$). Progression-free sets are often constructed "greedily," including every number so long as it is not in progression with any of the previous elements. Rankin characterized the greedy geometric-progression-free set in terms of the greedy arithmetic set. We characterize the greedy exponential set and prove that it has asymptotic density 1, and then discuss how the optimality of the greedy set depends on the family of functions used to define progressions. Traditionally, the size of a progression-free set is measured using the (upper) asymptotic density, however we consider several different notions of density, including the uniform and exponential densities.

math.NT↗

arXiv2015

The emergence of 4-cycles in polynomial maps over the extended integers

Let $f(x) \in \mathbb{Z}[x]$; for each integer $α$ it is interesting to consider the number of iterates $n_α$, if possible, needed to satisfy $f^{n_α}(α) = α$. The sets $\{α, f(α), \ldots, f^{n_α - 1}(α), α\}$ generated by the iterates of $f$ are called cycles. For $\mathbb{Z}[x]$ it is known that cycles of length 1 and 2 occur, and no others. While much is known for extensions to number fields, we concentrate on extending $\mathbb{Z}$ by adjoining reciprocals of primes. Let $\mathbb{Z}[1/p_1, \ldots, 1/p_n]$ denote $\mathbb{Z}$ extended by adding in the reciprocals of the $n$ primes $p_1, \ldots, p_n$ and all their products and powers with each other and the elements of $\mathbb{Z}$. Interestingly, cycles of length 4, called 4-cycles, emerge for polynomials in $\mathbb{Z}\left[1/p_1, \ldots, 1/p_n\right][x]$ under the appropriate conditions. The problem of finding criteria under which 4-cycles emerge is equivalent to determining how often a sum of four terms is zero, where the terms are $\pm 1$ times a product of elements from the list of $n$ primes. We investigate conditions on sets of primes under which 4-cycles emerge. We characterize when 4-cycles emerge if the set has one or two primes, and (assuming a generalization of the ABC conjecture) find conditions on sets of primes guaranteed not to cause 4-cycles to emerge.

math.NT↗

arXiv2015

Leading Digit Laws on Linear Lie Groups

We determine the leading digit laws for the matrix components of a linear Lie group $G$. These laws generalize the observations that the normalized Haar measure of the Lie group $\mathbb{R}^+$ is $dx/x$ and that the scale invariance of $dx/x$ implies the distribution of the digits follow Benford's law, which is the probability of observing a significand base $B$ of at most $s$ is $\log_B(s)$; thus the first digit is $d$ with probability $\log_B(1 + 1/d)$). Viewing this scale invariance as left invariance of Haar measure, we determine the power laws in significands from one matrix component of various such $G$. We also determine the leading digit distribution of a fixed number of components of a unit sphere, and find periodic behavior when the dimension of the sphere tends to infinity in a certain progression.

math.NT↗

arXiv2015

Gaps between zeros of GL(2) $L$-functions

Let $L(s,f)$ be an $L$-function associated to a primitive (holomorphic or Maass) cusp form $f$ on GL(2) over $\mathbb{Q}$. Combining mean-value estimates of Montgomery and Vaughan with a method of Ramachandra, we prove a formula for the mixed second moments of derivatives of $L(1/2+it,f)$ and, via a method of Hall, use it to show that there are infinitely many gaps between consecutive zeros of $L(s,f)$ along the critical line that are at least $\sqrt 3 = 1.732...$ times the average spacing. Using general pair correlation results due to Murty and Perelli in conjunction with a technique of Montgomery, we also prove the existence of small gaps between zeros of any primitive $L$-function of the Selberg class. In particular, when $f$ is a primitive holomorphic cusp form on GL(2) over $\mathbb{Q}$, we prove that there are infinitely many gaps between consecutive zeros of $L(s,f)$ along the critical line that are at most $< 0.823$ times the average spacing.

math.NT↗

arXiv2015

Equipartitions and a Distribution for Numbers: A Statistical Model for Benford's Law

A statistical model for the fragmentation of a conserved quantity is analyzed, using the principle of maximum entropy and the theory of partitions. Upper and lower bounds for the restricted partitioning problem are derived and applied to the distribution of fragments. The resulting power law directly leads to Benford's law for the first digits of the parts.

physics.data-an↗

arXiv2015

Spherical Matrix Ensembles

The spherical orthogonal, unitary, and symplectic ensembles (SOE/SUE/SSE) $S_β(N,r)$ consist of $N \times N$ real symmetric, complex hermitian, and quaternionic self-adjoint matrices of Frobenius norm $r$, made into a probability space with the uniform measure on the sphere. For each of these ensembles, we determine the joint eigenvalue distribution for each $N$, and we prove the empirical spectral measures rapidly converge to the semicircular distribution as $N \to \infty$. In the unitary case ($β=2$), we also find an explicit formula for the empirical spectral density for each $N$.

math.PR↗

arXiv2015

Gaussian Distribution of the Number of Summands in Generalized Zeckendorf Decompositions in Small Intervals

Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers ${F_n}$, with initial terms $F_1 = 1, F_2 = 2$. Previous work proved that as $n \to \infty$ the distribution of the number of summands in the Zeckendorf decompositions of $m \in [F_n, F_{n+1})$, appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in $[F_n, F_{n+1})$ share the same potential summands and hold for more general positive linear recurrence sequences $\{G_n\}$. We generalize these results to subintervals of $[G_n, G_{n+1})$ as $n \to \infty$ for certain sequences. The analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence $α(n) \to \infty$. As $n \to \infty$, for almost all $m \in [G_n, G_{n+1})$ the distribution of the number of summands in the generalized Zeckendorf decompositions of integers in the subintervals $[m, m + G_{α(n)})$, appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to $1$, $m$ has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and $[0, G_{α(n)})$ to obtain the result, since the summands are known to have Gaussian behavior in the latter interval.

math.NT↗