arXiv subjects
Publications and source records attributed to Aaron Berger.
Classical anti-concentration results focus on the random sum $S := \sum _{i=1}^n \xi_i v_i$, where $\xi_i$ are independent random variables and $v_i$ are real numbers. In this paper, we prove new concentration results concerning the random sum $S := \sum_{i=1}^n w_{\pi _i } v_i $, where $w_i , v_i$ are real numbers and $\pi$ is a random permutation.
We present a contact-based phenotyping robot platform that can autonomously insert nitrate sensors into cornstalks to proactively monitor macronutrient levels in crops. This task is challenging because inserting such sensors requires sub-centimeter precision in an environment which contains high levels of clutter, lighting variation, and occlusion. To address these challenges, we develop a robust perception-action pipeline to detect and grasp stalks, and create a custom robot gripper which mechanically aligns the sensor before inserting it into the stalk. Through experimental validation on 48 unique stalks in a cornfield in Iowa, we demonstrate our platform's capability of detecting a stalk with 94% success, grasping a stalk with 90% success, and inserting a sensor with 60% success. In addition to developing an autonomous phenotyping research platform, we share key challenges and insights obtained from deployment in the field. Our research platform is open-sourced, with additional information available at https://kantor-lab.github.io/cornbot.
The min-diameter of a directed graph $G$ is a measure of the largest distance between nodes. It is equal to the maximum min-distance $d_{min}(u,v)$ across all pairs $u,v \in V(G)$, where $d_{min}(u,v) = \min(d(u,v), d(v,u))$. Our work provides a $O(m^{1.426}n^{0.288})$-time $3/2$-approximation algorithm for min-diameter in DAGs, and a faster $O(m^{0.713}n)$-time almost-$3/2$-approximation variant. (An almost-$\alpha$-approximation algorithm determines the min-diameter to within a multiplicative factor of $\alpha$ plus constant additive error.) By a conditional lower bound result of [Abboud et al, SODA 2016], a better than $3/2$-approximation can't be achieved in truly subquadratic time under the Strong Exponential Time Hypothesis (SETH), so our result is conditionally tight. We additionally obtain a new conditional lower bound for min-diameter approximation in general directed graphs, showing that under SETH, one cannot achieve an approximation factor below 2 in truly subquadratic time. We also present the first study of approximating bichromatic min-diameter, which is the maximum min-distance between oppositely colored vertices in a 2-colored graph.
We study the following natural arithmetic question regarding intersecting families: how large can a family of subsets of integers from $\{1, \ldots n\}$ be such that, for every pair of subsets in the family, the intersection contains a sum $x + y = z$? We conjecture that any such sum-intersecting family must have size at most $\frac14 \cdot 2^{n}$ (which would be tight if correct). Towards this conjecture, we show that every sum-intersecting family has at most $0.32 \cdot 2^n$ subsets.
In this note we characterize when non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$-norm. We give a brief deduction of the fact that a bounded function on $\mathbb F_p^n$ with large $U^k$-norm must correlate with a classical polynomial when $k\leq p+1$. To the best of our knowledge, this result is new for $k=p+1$ (when $p>2$). We then prove that non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$-norm over $\mathbb F_p^n$ for all $k\geq p+2$, completely characterizing when classical polynomials suffice.
Ellis, Filmus, and Friedgut proved an old conjecture of Simonovits and S\'os showing that the maximum size of a triangle-intersecting family of graphs on $n$ vertices has size at most $2^{\binom{n}{2} - 3}$, with equality for the family of graphs containing some fixed triangle. They conjectured that their results extend to cross-intersecting families, as well to $K_t$-intersecting families. We prove these conjectures for $t \in \{3,4\}$, showing that if $\mathcal F_1$ and $\mathcal F_2$ are families of graphs on $n$ labeled vertices such that for any $G_1 \in \mathcal F_1$ and $G_2 \in \mathcal F_2$, $G_1 \cap G_2$ contains a $K_t$, then $\lvert \mathcal F_1 \rvert \lvert \mathcal F_2 \rvert \le 4^{\binom{n}{2} - \binom{t}{2}}$, with equality if and only if $\mathcal F_1 = \mathcal F_2$ consists of all graphs that contain some fixed $K_t$. We also establish a stability result. More generally, "$G_1 \cap G_2$ contains a $K_t$" can be replaced by "$G_1$ and $G_2$ agree on a non-$(t-1)$-colorable graph."
The following combinatorial conjecture arises naturally from recent ergodic-theoretic work of Ackelsberg, Bergelson, and Best. Let $M_1$, $M_2$ be $k\times k$ integer matrices, $G$ be a finite abelian group of order $N$, and $A\subseteq G^k$ with $|A|\ge\alpha N^k$. If $M_1$, $M_2$, $M_1-M_2$, and $M_1+M_2$ are automorphisms of $G^k$, is it true that there exists a popular difference $d \in G^k\setminus\{0\}$ such that \[\#\{x \in G^k: x, x+M_1d, x+M_2d, x+(M_1+M_2)d \in A\} \ge (\alpha^4-o(1))N^k.\] We show that this conjecture is false in general, but holds for $G = \mathbb{F}_p^n$ with $p$ an odd prime given the additional spectral condition that no pair of eigenvalues of $M_1M_2^{-1}$ (over $\overline{\mathbb{F}}_p$) are negatives of each other. In particular, the "rotated squares" pattern does not satisfy this eigenvalue condition, and we give a construction of a set of positive density in $(\mathbb{F}_5^n)^2$ for which that pattern has no nonzero popular difference. This is in surprising contrast to three-point patterns, which we handle over all compact abelian groups and which do not require an additional spectral condition.
We study the basic problem of assigning memoryless workers to tasks with dynamically changing demands. Given a set of $w$ workers and a multiset $T \subseteq[t]$ of $|T|=w$ tasks, a memoryless worker-task assignment function is any function $\phi$ that assigns the workers $[w]$ to the tasks $T$ based only on the current value of $T$. The assignment function $\phi$ is said to have switching cost at most $k$ if, for every task multiset $T$, changing the contents of $T$ by one task changes $\phi(T)$ by at most $k$ worker assignments. The goal of memoryless worker task assignment is to construct an assignment function with the smallest possible switching cost. In past work, the problem of determining the optimal switching cost has been posed as an open question. There are no known sub-linear upper bounds, and after considerable effort, the best known lower bound remains 4 (ICALP 2020). We show that it is possible to achieve polylogarithmic switching cost. We give a construction via the probabilistic method that achieves switching cost $O(\log w \log (wt))$ and an explicit construction that achieves switching cost $\operatorname{polylog} (wt)$. We also prove a super-constant lower bound on switching cost: we show that for any value of $w$, there exists a value of $t$ for which the optimal switching cost is $w$. Thus it is not possible to achieve a switching cost that is sublinear strictly as a function of $w$. Finally, we present an application of the worker-task assignment problem to a metric embeddings problem. In particular, we use our results to give the first low-distortion embedding from sparse binary vectors into low-dimensional Hamming space.
For a compact abelian group $G$, a corner in $G \times G$ is a triple of points $(x,y)$, $(x,y+d)$, $(x+d,y)$. The classical corners theorem of Ajtai and Szemer\'edi implies that for every $\alpha > 0$, there is some $\delta > 0$ such that every subset $A \subset G \times G$ of density $\alpha$ contains a $\delta$ fraction of all corners in $G \times G$, as $x,y,d$ range over $G$. Recently, Mandache proved a "popular differences" version of this result in the finite field case $G = \mathbb F_p^n$, showing that for any subset $A \subset G \times G$ of density $\alpha$, one can fix $d \neq 0$ such that $A$ contains a large fraction, now known to be approximately $\alpha^4$, of all corners with difference $d$, as $x,y$ vary over $G$. We generalize Mandache's result to all compact abelian groups $G$, as well as the case of corners in $\mathbb Z^2$.
Fici, Restivo, Silva, and Zamboni define a $\textit{$k$-anti-power}$ to be a concatenation of $k$ consecutive words that are pairwise distinct and have the same length. They ask for the maximum $k$ such that every aperiodic recurrent word must contain a $k$-anti-power, and they prove that this maximum must be 3, 4, or 5. We resolve this question by demonstrating that the maximum is 5. We also conjecture that if $W$ is a reasonably nice aperiodic morphic word, then there is some constant $C = C(W)$ such that for all $i,k\geq 1$, $W$ contains a $k$-anti-power with blocks of length at most $Ck$ beginning at its $i^\text{th}$ position. We settle this conjecture for binary words that are generated by a uniform morphism, characterizing the small exceptional set of words for which such a constant cannot be found. This generalizes recent results of the second author, Gaetz, and Narayanan that have been proven for the Thue-Morse word, which also show that such a linear bound is the best one can hope for in general.
We study tree-indexed random walks as introduced by Benjamini, Häggström, and Mossel, i.e. labelings of a tree for which adjacent vertices have labels differing by 1. It is a conjecture of those authors that the distribution of the range for any such tree is dominated by that of a path on the same number of edges. The two main variants of this conjecture considered in the literature are the $\textit{standard}$ walks, in which adjacent vertices must have labels differing by $\textit{exactly}$ 1, and $\textit{lazy}$ walks, in which adjacent vertices must have labels differing by $\textit{at most}$ 1. We confirm this conjecture for all trees in the lazy case and provide some partial results in the standard case.
For a graph property $\mathcal{P}$ and a common vertex set $V = \{1, 2, \ldots, n\}$, a family of graphs on $V$ is \emph{$\mathcal{P}$-intersecting} iff $G \cap H$ satisfies $\mathcal{P}$ for all $G,H$ in the family. Addressing a question of Chung, Graham, Frankl, and Shearer, we explore---for various $\mathcal{P}$---the maximum cardinality among all $\mathcal{P}$-intersecting families of graphs. In the connected-intersecting case, we resolve the question completely by a short linear algebraic proof showing this maximum is attained by taking all graphs containing a fixed spanning tree (though we show other extremal constructions as well). We also present a new lower bound for containing unions of a fixed subgraph.
For an abelian group $G$ and an integer $t > 0$, the \emph{modified Erdös--Ginzburg--Ziv constant} $s_t'(G)$ is the smallest integer $\ell$ such that any zero-sum sequence of length at least $\ell$ with elements in $G$ contains a zero-sum subsequence (not necessarily consecutive) of length $t$. We compute $s_t'(G)$ for $G = \mathbb Z/n\mathbb Z$ and for $t = n$, $G = (\mathbb Z/n\mathbb Z)^2$.
Let $\mathcal S$ be a multiset of integers. We say $\mathcal S$ is a $\textit{zero-sum sequence}$ if the sum of its elements is 0. We study zero-sum sequences whose elements lie in the interval $[-k,k]$ such that no subsequence of length $t$ is also zero-sum. Given these restrictions, Augspurger, Minter, Shoukry, Sissokho, Voss show that there are arbitrarily long $t$-avoiding, $k$-bounded zero-sum sequences unless $t$ is divisible by $\mathrm{LCM}(2,3,4,\dots,2k-1)$. We confirm a conjecture of these authors that for $k$ and $t$ such that this divisibility condition holds, every zero-sum sequence of length at least $t+k^2-k$ contains a zero-sum subsequence of length $t$, and that this is the minimal length for which this property holds.
Given a complex number $c$, define the divisor function $σ_c:\mathbb N\to\mathbb C$ by $σ_c(n)=\sum_{d\mid n}d^c$. In this paper, we look at $\overline{σ_{-r}(\mathbb N)}$, the topological closures of the image of $σ_{-r}$, when $r>1$. We exhibit new lower bounds on the number of connected components of $\overline{σ_{-r}(\mathbb N)}$, bringing this bound from linear in $r$ to exponential. Finally, we discuss the general structure of gaps of $\overline{σ_{-r}(\mathbb N)}$ in order to work towards a possible monotonicity result.
We study variants of Mastermind, a popular board game in which the objective is sequence reconstruction. In this two-player game, the so-called \textit{codemaker} constructs a hidden sequence $H = (h_1, h_2, \ldots, h_n)$ of colors selected from an alphabet $\mathcal{A} = \{1,2,\ldots, k\}$ (\textit{i.e.,} $h_i\in\mathcal{A}$ for all $i\in\{1,2,\ldots, n\}$). The game then proceeds in turns, each of which consists of two parts: in turn $t$, the second player (the \textit{codebreaker}) first submits a query sequence $Q_t = (q_1, q_2, \ldots, q_n)$ with $q_i\in \mathcal{A}$ for all $i$, and second receives feedback $Δ(Q_t, H)$, where $Δ$ is some agreed-upon function of distance between two sequences with $n$ components. The game terminates when $Q_t = H$, and the codebreaker seeks to end the game in as few turns as possible. Throughout we let $f(n,k)$ denote the smallest integer such that the codebreaker can determine any $H$ in $f(n,k)$ turns. We prove three main results: First, when $H$ is known to be a permutation of $\{1,2,\ldots, n\}$, we prove that $f(n, n)\ge n - \log\log n$ for all sufficiently large $n$. Second, we show that Knuth's Minimax algorithm identifies any $H$ in at most $nk$ queries. Third, when feedback is not received until all queries have been submitted, we show that $f(n,k)=Ω(n\log k)$.
A $\textit{ladder}$ is a set $S \subseteq \mathbb Z^+$ such that any finite coloring of $\mathbb Z$ contains arbitrarily long monochromatic progressions with common difference in $S$. Van der Waerden's theorem famously asserts that $\mathbb Z^+$ itself is a ladder. We also discuss variants of ladders, namely $\textit{accessible}$ and $\textit{walkable}$ sets, which are sets $S$ such that any coloring of $\mathbb Z$ contains arbitrarily long (for accessible sets) or infinite (for walkable sets) monochromatic sequences with consecutive differences in $S$. We show that sets with upper density 1 are ladders and walkable. We also show that all directed graphs with infinite chromatic number are accessible, and reduce the bound on the walkability order of sparse sets from 3 to 2, making it tight.