Related papers: The lonely runner conjecture holds for eight runne…

In this paper, we study the distribution of the boundary points of expansion. As an application, we say something about the lonely runner problem. We show that given $k$ runners $\mathcal{S}_i$ round a unit circular track with the condition…

In this note we develop a linear programming framework to produce upper and lower bounds for the lonely runner problem.

Given collections A and B of residue classes modulo m and n, respectively, we investigate conditions on A and B that ensure that, for at least some a in A and b in B, the linear system x = a mod m, x = b mod n has an integer solution, and…

Let $||x||$ be the absolute distance from $x$ to the nearest integer. For a set of distinct positive integral speeds $v_1, \ldots, v_n$, we define its maximum loneliness, also known as the gap $\delta$, to be $$ML(v_1,\ldots,v_n) = \max_{t…

A fixed number of passengers independently board one of several buses uniformly at random. The lonely passenger problem is to prove that the probability of at least one passenger being the only one in their bus is increasing in the number…

Consider the circle $C$ of length 1 and a circular arc $A$ of length $\ell\in (0,1)$. It is shown that there exists $k=k(\ell) \in \mathbb{N}$, and a schedule for $k$ runners along the circle with $k$ constant but distinct positive speeds…

We are concerned with the computational problem of determining the covering radius of a rational polytope. This parameter is defined as the minimal dilation factor that is needed for the lattice translates of the correspondingly dilated…

Levin's conjecture has been established to hold true for group equations of length up to seven. Recently, it is shown that Levin's conjecture is also true (modulo exceptional cases) for some group equations of length eight and nine. In this…

We prove that deciding whether the Runner can win this turn (mate-in-1) in the Netrunner card game generalized to allow decks to contain an arbitrary number of copies of a card is weakly NP-hard. We also prove that deciding whether the Corp…

We show that the Coleman-Oort conjecture can be reduced to three particular cases. As an application we extend a result of Lu and Zuo, to the effect that for g at least 8 the Coleman-Oort conjecture is true on the hyperelliptic locus.

We prove that there is a matching between 2 intervals of positive integers of the same even length, with corresponding pairs coprime, provided the intervals are in $[n]$ and their lengths are $>c(\log n)^2$, for a positive constant $c$.…

This paper presents a sharp approximation of the density of long runs of a random walk conditioned on its end value or by an average of a functions of its summands as their number tends to infinity. The conditioning event is of moderate or…

We prove the explicit formula for the probability of a run of r successes in n trials.

Eight combinatorial identities are listed and proved by counting paths in the one-dimensional random walk. Four of these identities are assumed to be new.

In forecasting competitions, the traditional mechanism scores the predictions of each contestant against the outcome of each event, and the contestant with the highest total score wins. While it is well-known that this traditional mechanism…

Physical activity is known to help improve and maintain one's health. In particular, recreational running has become increasingly popular in recent years. Yet, lack of motivation often interferes with people's routines and thus may prohibit…

Consider a walk in the plane made of $n$ unit steps, with directions chosen independently and uniformly at random at each step. Rayleigh's theorem asserts that the probability for such a walk to end at a distance less than 1 from its…

LLM leaderboards often rely on single stochastic runs, but how many repetitions are required for reliable conclusions remains unclear. We re-evaluate eight state-of-the-art models on the AI4Math Benchmark with three independent runs per…

We consider transient nearest neighbor random walks on the positive part of the real line. We give criteria for the finiteness of the number of cutpoints and strong cutpoints. Examples and open problems are presented.

We prove Dejean's conjecture. Specifically, we show that Dejean's conjecture holds for the last remaining open values of n, namely 15 <= n <= 26.