arXiv subjects

Tom Johnston

Publications and source records attributed to Tom Johnston.

23 records · Page 2Linked to original sources

arXiv2021

Lipschitz bijections between boolean functions

We answer four questions from a recent paper of Rao and Shinkar on Lipschitz bijections between functions from $\{0,1\}^n$ to $\{0,1\}$. (1) We show that there is no $O(1)$-bi-Lipschitz bijection from $\mathrm{Dictator}$ to $\mathrm{XOR}$ such that each output bit depends on $O(1)$ input bits. (2) We give a construction for a mapping from $\mathrm{XOR}$ to $\mathrm{Majority}$ which has average stretch $O(\sqrt{n})$, matching a previously known lower bound. (3) We give a 3-Lipschitz embedding $ϕ: \{0,1\}^n \to \{0,1\}^{2n+1}$ such that $\mathrm{XOR}(x) = \mathrm{Majority}(ϕ(x))$ for all $x \in \{0,1\}^n$. (4) We show that with high probability there is a $O(1)$-bi-Lipschitz mapping from $\mathrm{Dictator}$ to a uniformly random balanced function.

math.CO↗

arXiv2021

Exact hyperplane covers for subsets of the hypercube

Alon and Füredi (1993) showed that the number of hyperplanes required to cover $\{0,1\}^n\setminus \{0\}$ without covering $0$ is $n$. We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the hypercube. In particular, we provide exact solutions for covering $\{0,1\}^n$ while missing up to four points and give asymptotic bounds in the general case. Several interesting questions are left open.

math.CO↗

arXiv2021

Cyclically covering subspaces in $\mathbb{F}_2^n$

A subspace of $\mathbb{F}_2^n$ is called cyclically covering if every vector in $\mathbb{F}_2^n$ has a cyclic shift which is inside the subspace. Let $h_2(n)$ denote the largest possible codimension of a cyclically covering subspace of $\mathbb{F}_2^n$. We show that $h_2(p)= 2$ for every prime $p$ such that 2 is a primitive root modulo $p$, which, assuming Artin's conjecture, answers a question of Peter Cameron from 1991. We also prove various bounds on $h_2(ab)$ depending on $h_2(a)$ and $h_2(b)$ and extend some of our results to a more general set-up proposed by Cameron, Ellis and Raynaud.

math.CO↗

arXiv2018

Intersection sizes of linear subspaces with the hypercube

We continue the study by Melo and Winter [arXiv:1712.01763, 2017] on the possible intersection sizes of a $k$-dimensional subspace with the vertices of the $n$-dimensional hypercube in Euclidean space. Melo and Winter conjectured that all intersection sizes larger than $2^{k-1}$ (the "large" sizes) are of the form $2^{k-1}+2^i$. We show that this is almost true: the large intersection sizes are either of this form or of the form $35\cdot 2^{k-6}$. We also disprove a second conjecture of Melo and Winter by proving that a positive fraction of the "small" values is missing.

math.CO↗

arXiv2018

Exceptional graphs for the random walk

If $\mathcal{W}$ is the simple random walk on the square lattice $\mathbb{Z}^2$, then $\mathcal{W}$ induces a random walk $\mathcal{W}_G$ on any spanning subgraph $G\subset \mathbb{Z}^2$ of the lattice as follows: viewing $\mathcal{W}$ as a uniformly random infinite word on the alphabet $\{\mathbf{x}, -\mathbf{x}, \mathbf{y}, -\mathbf{y} \}$, the walk $\mathcal{W}_G$ starts at the origin and follows the directions specified by $\mathcal{W}$, only accepting steps of $\mathcal{W}$ along which the walk $\mathcal{W}_G$ does not exit $G$. For any fixed subgraph $G \subset \mathbb{Z}^2$, the walk $\mathcal{W}_G$ is distributed as the simple random walk on $G$, and hence $\mathcal{W}_G$ is almost surely recurrent in the sense that $\mathcal{W}_G$ visits every site reachable from the origin in $G$ infinitely often. This fact naturally leads us to ask the following: does $\mathcal{W}$ almost surely have the property that $\mathcal{W}_G$ is recurrent for \emph{every} subgraph $G \subset \mathbb{Z}^2$? We answer this question negatively, demonstrating that exceptional subgraphs exist almost surely. In fact, we show more to be true: exceptional subgraphs continue to exist almost surely for a countable collection of independent simple random walks, but on the other hand, there are almost surely no exceptional subgraphs for a branching random walk.

math.PR↗