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Zachary Chase

Publications and source records attributed to Zachary Chase.

25 records · Page 2Linked to original sources
arXiv2021

Additive Conjugacy and the Bohr Compactification of Orthogonal Representations

We say that two unitary or orthogonal representations of a finitely generated group $G$ are additive conjugates if they are intertwined by an additive map, which need not be continuous. We associate to each representation of $G$ a topological action that is a complete additive conjugacy invariant: the action of $G$ by group automorphisms on the Bohr compactification of the underlying Hilbert space. Using this construction we show that the property of having almost invariant vectors is an additive conjugacy invariant. As an application we show that $G$ is amenable if and only if there is a nonzero homomorphism from $L^2(G)$ into $\mathbb{R}/\mathbb{Z}$ that is invariant to the $G$-action.

math.GR↗
arXiv2020

New Lower Bounds for Trace Reconstruction

We improve the lower bound on worst case trace reconstruction from $Ω\left(\frac{n^{5/4}}{\sqrt{\log n}}\right)$ to $Ω\left(\frac{n^{3/2}}{\log^{7} n}\right)$. As a consequence, we improve the lower bound on average case trace reconstruction from $Ω\left(\frac{\log^{9/4}n}{\sqrt{\log\log n}}\right)$ to $Ω\left(\frac{\log^{5/2}n}{(\log\log n)^{7}}\right)$.

math.PR↗
arXiv2020

On the Iterates of Digit Maps

Given a base $b$, a "digit map" is a map $f: \mathbb{Z}^{\ge 0} \to \mathbb{Z}^{\ge 0}$ of the form $f(\sum_{i=0}^n a_ib^i) = \sum_{i=0}^n f_*(a_i)$, $0 \le a_i \le b-1$ for each $i$, where $f_* : \{0,1,\dots, b-1\} \to \mathbb{Z}^{\ge 0}$ satisfies $f_*(0) = 0$ and $f_*(1) = 1$. It has been proven for $b=10$ and $f_*(m) = m^2$, and various generalizations thereof, that there are arbitrarily long sequences of consecutive positive integers that end up at $1$ under repeated application of $f$. In this paper, we significantly generalize these results, providing a complete classification of digit maps for which, given any periodic point $n$, there are arbitrarily long sequences of consecutive positive integers that end up $n$.

math.NT↗
arXiv2018

The Maximum Number of Three Term Arithmetic Progressions, and Triangles in Cayley Graphs

Let $G$ be a finite Abelian group. For a subset $S \subseteq G$, let $T_3(S)$ denote the number of length three arithemtic progressions in $S$ and Prob[$S$] $= \frac{1}{|S|^2}\sum_{x,y \in S} 1_S(x+y)$. For any $q \ge 1$ and $α\in [0,1]$, and any $S \subseteq G$ with $|S| = \frac{|G|}{q+α}$, we show $\frac{T_3(S)}{|S|^2}$ and Prob[$S$] are bounded above by $\max\left(\frac{q^2-αq+α^2}{q^2},\frac{q^2+2αq+4α^2-6α+3}{(q+1)^2},γ_0\right)$, where $γ_0 < 1$ is an absolute constant. As a consequence, we verify a graph theoretic conjecture of Gan, Loh, and Sudakov for Cayley graphs.

math.CO↗
arXiv2018

Learning Time Dependent Choice

We explore questions dealing with the learnability of models of choice over time. We present a large class of preference models defined by a structural criterion for which we are able to obtain an exponential improvement over previously known learning bounds for more general preference models. This in particular implies that the three most important discounted utility models of intertemporal choice -- exponential, hyperbolic, and quasi-hyperbolic discounting -- are learnable in the PAC setting with VC dimension that grows logarithmically in the number of time periods. We also examine these models in the framework of active learning. We find that the commonly studied stream-based setting is in general difficult to analyze for preference models, but we provide a redeeming situation in which the learner can indeed improve upon the guarantees provided by PAC learning. In contrast to the stream-based setting, we show that if the learner is given full power over the data he learns from -- in the form of learning via membership queries -- even very naive algorithms significantly outperform the guarantees provided by higher level active learning algorithms.

cs.GT↗