- Bronfman Science Center
Williams College
Williamstown, MA 01267 - 413-597-3293
Steven J. Miller
Williams College, Mathematics and Statistics, Faculty Member
- My main research interests are in number theory (especially random matrix theory, elliptic curves and additive and co... moreMy main research interests are in number theory (especially random matrix theory, elliptic curves and additive and computational number theory), probability and statistics (especially Benford's law, linear programming and sabermetrics). I'm married with two kids, and I enjoyed sailing, reading and tennis when I had the time.
I have written or refereed articles in the following fields (in addition to mathematics): accounting, biology, economics, geology, marketing, sabermetrics and statistics, and am always looking for interesting projects.
I have written a book (with Ramin Takloo-Bighash), "An Invitation to Modern Number Theory" (Princeton University Press), whose purpose is to introduce undergraduates and beginning graduate students to modern number theory (either as a standard textbook, or through suggested research problems). The book's homepage is: http://www.williams.edu/go/math/sjmiller/public_html/book/index.htmledit
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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... more
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 d...
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This paper describes a model that generates weekly movie schedules in a multiplex movie theater. A movie schedule specifies within each day of the week, on which screen(s) different movies will be played, and at which time(s). The model... more
This paper describes a model that generates weekly movie schedules in a multiplex movie theater. A movie schedule specifies within each day of the week, on which screen(s) different movies will be played, and at which time(s). The model consists of two parts: (i) conditional forecasts of the number of visitors per show for any possible starting time; and (ii) an optimization procedure that quickly finds an almost optimal schedule (which can be demonstrated to be close to the optimal schedule). To generate this schedule we formulate the so-called movie scheduling problem as a generalized set partitioning problem. The latter is solved with an algorithm based on column generation techniques. We have applied this combined demand forecasting /schedule optimization procedure to a multiplex in Amsterdam where we supported the scheduling of fourteen movie weeks. The proposed model not 2 only makes movie scheduling easier and less time consuming, but also generates schedules that would attra...
Zeckendorf proved that every positive integer $n$ can be written uniquely as the sum of non-adjacent Fibonacci numbers; a similar result, though with a different notion of a legal decomposition, holds for many other sequences. We use... more
Zeckendorf proved that every positive integer $n$ can be written uniquely as the sum of non-adjacent Fibonacci numbers; a similar result, though with a different notion of a legal decomposition, holds for many other sequences. We use these decompositions to construct a two-player game, which can be completely analyzed for linear recurrence relations of the form $G_n = \sum_{i=1}^{k} c G_{n-i}$ for a fixed positive integer $c$ ($c=k-1=1$ gives the Fibonaccis). Given a fixed integer $n$ and an initial decomposition of $n = n G_1$, the two players alternate by using moves related to the recurrence relation, and whomever moves last wins. The game always terminates in the Zeckendorf decomposition, though depending on the choice of moves the length of the game and the winner can vary. We find upper and lower bounds on the number of moves possible; for the Fibonacci game the upper bound is on the order of $n\log n$, and for other games we obtain a bound growing linearly with $n$. For the F...
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A set $A$ is MSTD (more-sum-than-difference) if $|A+A|>|A-A|$. Though MSTD sets are rare, Martin and O'Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of $\{1,2,\ldots ,r\}$ as... more
A set $A$ is MSTD (more-sum-than-difference) if $|A+A|>|A-A|$. Though MSTD sets are rare, Martin and O'Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of $\{1,2,\ldots ,r\}$ as $r\rightarrow\infty$. Later, Asada et al. showed that there exists a positive constant lower bound for the proportion of decompositions of $\{1,2,\ldots,r\}$ into two MSTD subsets as $r\rightarrow\infty$. However, the method is probabilistic and does not give explicit decompositions. Continuing this work, we provide an efficient method to partition $\{1,2,\ldots,r\}$ (for $r$ sufficiently large) into $k \ge 2$ MSTD subsets, positively answering a question raised by Asada et al. as to whether this is possible for all such $k$. Next, let $R(k)$ be the smallest integer such that for all $r\ge R(k)$, $\{1,2,\ldots,r\}$ can be $k$-decomposed into MSTD subsets. We establish rough lower and upper bounds for $R(k)$. Lastly, we provide a sufficient condition on...
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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... more
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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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 ≤ i ≤ n− 1 there is a distance that occurs exactly i times. For small n this is possible... more
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 ≤ i ≤ 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 ≤ 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.
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A finite set of integers $A$ is a sum-dominant (also called an More Sums Than Differences or MSTD) set if $|A+A| > |A-A|$. While almost all subsets of $\{0, \dots, n\}$ are not sum-dominant, interestingly a small positive percentage... more
A finite set of integers $A$ is a sum-dominant (also called an More Sums Than Differences or MSTD) set if $|A+A| > |A-A|$. While almost all subsets of $\{0, \dots, n\}$ are not sum-dominant, interestingly a small positive percentage are. We explore sufficient conditions on infinite sets of positive integers such that there are either no sum-dominant subsets, at most finitely many sum-dominant subsets, or infinitely many sum-dominant subsets. In particular, we prove no subset of the Fibonacci numbers is a sum-dominant set, establish conditions such that solutions to a recurrence relation have only finitely many sum-dominant subsets, and show there are infinitely many sum-dominant subsets of the primes.
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Edouard Zeckendorf [5] proved that every positive integer n can be uniquely written as the sum of nonadjacent Fibonacci numbers, known as the Zeckendorf decomposition. Based on Zeckendorf’s decomposi-tion, we have the Zeckendorf game for... more
Edouard Zeckendorf [5] proved that every positive integer n can be uniquely written as the sum of nonadjacent Fibonacci numbers, known as the Zeckendorf decomposition. Based on Zeckendorf’s decomposi-tion, we have the Zeckendorf game for multiple players. We show that when the Zeckendorf game has at least three players, none of the play-ers have a winning strategy for n ≥ 5. Then we extend the multiplayer game to the multialliance 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.
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A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers {Fn}n=1; Lekkerkerker proved that the average number of summands for integers in [Fn,Fn+1) is n/(φ2 + 1),... more
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers {Fn}n=1; Lekkerkerker proved that the average number of summands for integers in [Fn,Fn+1) is n/(φ2 + 1), with φ the golden mean. This has been generalized to certain classes of linear recurrence relations, where using techniques from number theory and ergodic theory the fluctuations about the mean are shown to be Gaussian. We discuss an alternative proof that is more combinatorial, and comment on generalizations to related decompositions and problems which have not been handled by these other methods. For example, every integer can be written uniquely as a sum of the ±Fn’s, such that every two terms of the same (opposite) sign differ in index by at least 4 (3). The distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely−(21−2φ)/(29+2φ)≈−0.551058. Another consequence of th...
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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... more
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...
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In this paper, we define a [Formula: see text]-Diophantine [Formula: see text]-tuple to be a set of [Formula: see text] positive integers such that the product of any [Formula: see text] distinct positive integers is one less than a... more
In this paper, we define a [Formula: see text]-Diophantine [Formula: see text]-tuple to be a set of [Formula: see text] positive integers such that the product of any [Formula: see text] distinct positive integers is one less than a perfect square. We study these sets in finite fields [Formula: see text] for odd prime [Formula: see text] and guarantee the existence of a [Formula: see text]-Diophantine [Formula: see text]-tuple provided [Formula: see text] is larger than some explicit lower bound. We also give a formula for the number of 3-Diophantine triples in [Formula: see text] as well as an asymptotic formula for the number of [Formula: see text]-Diophantine [Formula: see text]-tuples.
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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... more
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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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... more
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 sequen...
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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... more
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 $Z(n)$ i...
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A finite set of integersA is a sum-dominant (also called a More Sums Than Differences or MSTD) set if |A + A| > |A − A|. While almost all subsets of {0, . . . , n} are not sum-dominant, interestingly a small positive perce ntage are.... more
A finite set of integersA is a sum-dominant (also called a More Sums Than Differences or MSTD) set if |A + A| > |A − A|. While almost all subsets of {0, . . . , n} are not sum-dominant, interestingly a small positive perce ntage are. We explore sufficient conditions on infinite sets of positive inte gers such that there are either no sum-dominant subsets, at most finitely many sum-dominant subsets, or infinitely many sum-dominant subsets. In particular, we prove no subse t of the Fibonacci numbers is a sum-dominant set, establish conditions such that s olutions to a recurrence relation have only finitely many sum-dominant subsets, and s how there are infinitely many sum-dominant subsets of the primes.
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An interesting question known as the Gaussian Moat problem asks whether it is possible to walk to infinity along the Gaussian primes with a bounded step size. We examine a similar version of this problem in the real quadratic integer ring... more
An interesting question known as the Gaussian Moat problem asks whether it is possible to walk to infinity along the Gaussian primes with a bounded step size. We examine a similar version of this problem in the real quadratic integer ring Z[ √ 2] whose primes mostly cluster along the asymptotes y = ±x/ √ 2 as compared to the Gaussian primes, which mainly cluster at the origin. A probabilistic model of primes a + b √ 2 in Z[ √ 2] is then constructed according to their norms a− 2b by applying the Prime Number Theorem and a combinatorial theorem for counting the number of lattice points in the region |a− 2b| ≤ n. Lastly, we perform a few moat calculations in Z[ √ 2] for various step sizes and make a conjecture about the existence of a prime walk to infinity.