Carsten Peterson’s research while affiliated with University of Michigan and other places

A more sums than differences (MSTD) set A is a subset of $\mathbb{Z}$ for which $|A+A| > |A-A|$. Martin and O'Bryant used probabilistic techniques to prove that a non-vanishing proportion of subsets of $\{1, \dots, n\}$ are MSTD as $n \to \infty$. However, to date only a handful of explicit constructions of MSTD sets are known. We study finite collections of disjoint intervals on the real line, $\mathbb{I}$, and explore the MSTD question for such sets, as well as the relation between such sets and MSTD subsets of $\mathbb{Z}$. In particular we show that every finite subset of $\mathbb{Z}$ can be transformed into an element of $\mathbb{I}$ with the same additive behavior. Using tools from discrete geometry, we show that there are no MSTD sets in $\mathbb{I}$ consisting of three or fewer intervals, but there are MSTD sets for four or more intervals. Furthermore, we show how to obtain an infinite parametrized family of MSTD subsets of $\mathbb{Z}$ from a single such set A; these sets are parametrized by lattice points satisfying simple congruence relations contained in a polyhedral cone associated to A.

Given a recurrence sequence H, with Hn= c1Hn-1+ ⋯ + ctHn-t where ci∈ N0 for all i and c1, ct≥ 1 , the generalized Zeckendorf decomposition (gzd) of m∈ N0 is the unique representation of m using H composed of blocks lexicographically less than σ= (c1, ⋯ , ct). We prove that the gzd of m uses the fewest number of summands among all representations of m using H, for all m, if and only if σ is weakly decreasing. We develop an algorithm for moving from any representation of m to the gzd, the analysis of which proves that σ weakly decreasing implies summand minimality. We prove that the gzds of numbers of the form v0Hn+ ⋯ + vℓHn-ℓ converge in a suitable sense as n→ ∞; furthermore we classify three distinct behaviors for this convergence. We use this result, together with the irreducibility of certain families of polynomials, to exhibit a representation with fewer summands than the gzd if σ is not weakly decreasing.

A more sums than differences (MSTD) set A is a subset of $\mathbb{Z}$ for which $|A+A| > |A-A|$. Martin and O'Bryant used probabilistic techniques to prove that a non-vanishing proportion of subsets of $\{1, \dots, n\}$ are MSTD as $n \to \infty$. However, to date only a handful of explicit constructions of MSTD sets are known. We study finite collections of disjoint intervals on the real line, $\mathbb{I}$, and explore the MSTD question for such sets, as well as the relation between such sets and MSTD subsets of $\mathbb{Z}$. In particular we show that every finite subset of $\mathbb{Z}$ can be transformed into an element of $\mathbb{I}$ with the same additive behavior. Using tools from discrete geometry, we show that there are no MSTD sets in $\mathbb{I}$ consisting of three or fewer intervals, but there are MSTD sets for four or more intervals. Furthermore, we show how to obtain an infinite parametrized family of MSTD subsets of $\mathbb{Z}$ from a single such set A; these sets are parametrized by lattice points satisfying simple congruence relations contained in a polyhedral cone associated to A.

A bidirectional ballot sequence (BBS) is a finite binary sequence with the property that every prefix and suffix contains strictly more ones than zeros. BBSs were introduced by Zhao, and independently by Bosquet-M\'elou and Ponty as (1,1)-culminating paths. Both sets of authors noted the difficulty in counting these objects, and to date research on bidirectional ballot sequences has been concerned with asymptotics. We introduce a continuous analogue of bidirectional ballot sequences which we call bidirectional gerrymanders, and show that the set of bidirectional gerrymanders form a convex polytope sitting inside the unit cube, which we refer to as the bidirectional ballot polytope. We prove that every $(2n-1)$-dimensional unit cube can be partitioned into $2n-1$ isometric copies of the $(2n-1)$-dimensional bidirectional ballot polytope. Furthermore, we show that the vertices of this polytope are all also vertices of the cube, and that the vertices are in bijection with BBSs. An immediate corollary is a geometric explanation of the result of Zhao and of Bosquet-M\'elou and Ponty that the number of BBSs of length n is $\Theta(2^n/n)$.

Zeckendorf proved that every number can be uniquely represented as a sum of non-consecutive Fibonacci numbers. This has been extended in many ways, including to linear recurrences $H_n=c_1 H_{n-1} + \cdots + c_t H_{n-t}$ where the $c_i$ are non-negative integers and $c_1$, $c_t \ge 1$. Every number has a unique generalized Zeckendorf decomposition (gzd) -- a representation composed of blocks that are lexicographically less than $(c_1,\dots,c_t)$, which we call the signature. We prove that the gzd of a positive integer m uses the fewest number of summands out of all representations for m using the same recurrence sequence, for all m, if and only if the signature of the linear recurrence is weakly decreasing (i.e., $c_1 \ge \cdots \ge c_t$). Following the parallel with well-known base d representations, we develop a framework for naturally moving between representations of the same number using a linear recurrence, which we then utilize to construct an algorithm to turn any representation of a number into the gzd. To prove sufficiency, we show that if the signature is weakly decreasing then our algorithm results in fewer summands. To prove necessity we proceed by divide and conquer, breaking the analysis into several cases. When $c_1 > 1$, we give an example of a non-gzd representation of a number and show that it has fewer summands than the gzd by performing the same above-mentioned algorithm. When $c_1 = 1$, we non-constructively prove the existence of a counterexample by utilizing the irreducibility of a certain family of polynomials together with growth rate arguments.