The lonely runner conjecture, now over fifty years old, concerns the following problem. On a unit-length circular track, consider $m$ runners starting at the same time and place, each runner having a different constant speed. The conjecture asserts that each runner is lonely at some point in time, meaning at a distance at least $1/m$ from the others. We formulate a function field analogue, and give a positive answer in some cases in the new setting.

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$ if for every $\unicode[STIX]{x1D716}>0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less than $1+\unicode[STIX]{x1D716}$. We show that almost isometric embeddability conditions are equal for the following classes of spaces.

1. (a) Quotients of Euclidean spaces by isometric actions of finite groups.

2. (b) $L_{2}$-Wasserstein spaces over Euclidean spaces.

3. (c) Compact flat manifolds.

4. (d) Compact flat orbifolds.

5. (e) Quotients of connected compact bi-invariant Lie groups by isometric actions of compact Lie groups. (This one is the most surprising.)

We call spaces which satisfy these conditions finite flat spaces. Since Markov-type constants depend only on finite subsets, we can conclude that connected compact bi-invariant Lie groups and their quotients have Markov type 2 with constant 1.

Motivated by the Erdős–Szekeres convex polytope conjecture in $\mathbb{R}^{d}$, we initiate the study of the following induced Ramsey problem for hypergraphs. Given integers $n>k\geqslant 5$, what is the minimum integer $g_{k}(n)$ such that any$k$-uniform hypergraph on $g_{k}(n)$ vertices with the property that any set of $k+1$ vertices induces 0, 2, or 4 edges, contains an independent set of size $n$. Our main result shows that $g_{k}(n)>2^{cn^{k-4}}$, where $c=c(k)$.

In this paper, we improve the moment estimates for the gaps between numbers that can be represented as a sum of two squares of integers. We consider a certain sum of Bessel functions and prove the upper bound for its mean value. This bound provides estimates for the $\unicode[STIX]{x1D6FE}$th moments of gaps for all $\unicode[STIX]{x1D6FE}\leqslant 2$.

We consider the following question: for which metrizable separable spaces $X$ does the free abelian topological group $A(X\oplus X)$ isomorphically embed into $A(X)$. While for many natural spaces $X$ such an embedding exists, our main result shows that if $X$ is a Cook continuum or $X$ is a rigid Bernstein set, then $A(X\oplus X)$ does not embed into $A(X)$ as a topological subgroup. The analogous statement is true for the free boolean group $B(X)$.

We prove that for every sufficiently large integer $n$, the polynomial $1+x+x^{2}/11+x^{3}/111+\cdots +x^{n}/111\ldots 1$ is irreducible over the rationals, where the coefficient of $x^{k}$ for $1\leqslant k\leqslant n$ is the reciprocal of the decimal number consisting of $k$ digits which are each $1$. Similar results following from the same techniques are discussed.

In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let $\ell$ be a prime, $q$ a prime power and consider the ensemble ${\mathcal{H}}_{g,\ell }$ of $\ell$-cyclic covers of $\mathbb{P}_{\mathbb{F}_{q}}^{1}$ of genus $g$. We assume that $q\not \equiv 0,1~\text{mod}~\ell$. If $2g+2\ell -2\not \equiv 0~\text{mod}~(\ell -1)\operatorname{ord}_{\ell }(q)$, then ${\mathcal{H}}_{g,\ell }$ is empty. Otherwise, the number of rational points on a random curve in ${\mathcal{H}}_{g,\ell }$ distributes as $\sum _{i=1}^{q+1}X_{i}$ as $g\rightarrow \infty$, where $X_{1},\ldots ,X_{q+1}$ are independent and identically distributed random variables taking the values $0$ and $\ell$ with probabilities $(\ell -1)/\ell$ and $1/\ell$, respectively. The novelty of our result is that it works in the absence of a primitive $\ell$th root of unity, the presence of which was crucial in previous studies.

We establish asymptotic formulae for the number of $k$-free values of square-free polynomials $F(x_{1},\ldots ,x_{n})\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ of degree $d\geqslant 2$ for any $n\geqslant 1$, including when the variables are prime, as long as $k\geqslant (3d+1)/4$. This generalizes a work of Browning.

We study Piatetski-Shapiro sequences $(\lfloor n^{c}\rfloor )_{n}$ modulo $m$, for non-integer $c>1$ and positive $m$, and we are particularly interested in subword occurrences in those sequences. We prove that each block $\in \{0,1\}^{k}$ of length $k<c+1$ occurs as a subword with the frequency $2^{-k}$, while there are always blocks that do not occur. In particular, those sequences are not normal. For $1<c<2$, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi–Kátai criterion, we prove that the sequence $\lfloor n^{c}\rfloor$ modulo $m$ is asymptotically orthogonal to multiplicative functions bounded by 1 and with mean value 0.

In this paper various analytic techniques are combined in order to study the average of a product of a Hecke $L$-function and a symmetric square $L$-function at the central point in the weight aspect. The evaluation of the second main term relies on the theory of Maaß forms of half-integral weight and the Rankin–Selberg method. The error terms are bounded using the Liouville–Green approximation.

We show that if a permutation $\unicode[STIX]{x1D70B}$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]$ of the interval $[1,\unicode[STIX]{x1D70B}]$ is zero. As a consequence, we prove that the proportion of permutations of length $n$ with principal Möbius function equal to zero is asymptotically bounded below by $(1-1/e)^{2}\geqslant 0.3995$. This is the first result determining the value of $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]$ for an asymptotically positive proportion of permutations $\unicode[STIX]{x1D70B}$. We further establish other general conditions on a permutation $\unicode[STIX]{x1D70B}$ that ensure $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]=0$, including the occurrence in $\unicode[STIX]{x1D70B}$ of any interval of the form $\unicode[STIX]{x1D6FC}\oplus 1\oplus \unicode[STIX]{x1D6FD}$.

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^{\ast }$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^{\ast }$-polynomial is $\unicode[STIX]{x1D6FE}$-positive and real-rooted. This proves Gal’s conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthening due to Nevo and Petersen [On $\unicode[STIX]{x1D6FE}$-vectors satisfying the Kruskal–Katona inequalities. Discrete Comput. Geom.45(3) (2011), 503–521].

For each $n\geqslant 2$ we construct a measurable subset of the unit ball in $\mathbb{R}^{n}$ that does not contain pairs of points at distance 1 and whose volume is greater than $(1/2)^{n}$ times the volume of the unit ball. This disproves a conjecture of Larman and Rogers from 1972.