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.

In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results. Except for parallelograms and centrally symmetric hexagons, there are no other convex domains that can form two-, three- or four-fold lattice tilings in the Euclidean plane. However, there are both octagons and decagons that can form five-fold lattice tilings. Whenever $n\geqslant 3$, there are non-parallelohedral polytopes that can form five-fold lattice tilings in the $n$-dimensional Euclidean space.

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

It was shown by Gruslys, Leader and Tan that any finite subset of $\mathbb{Z}^{n}$ tiles $\mathbb{Z}^{d}$ for some $d$. The first non-trivial case is the punctured interval, which consists of the interval $\{-k,\ldots ,k\}\subset \mathbb{Z}$ with its middle point removed: they showed that this tiles $\mathbb{Z}^{d}$ for $d=2k^{2}$, and they asked if the dimension needed tends to infinity with $k$. In this note we answer this question: we show that, perhaps surprisingly, every punctured interval tiles $\mathbb{Z}^{4}$.

We characterize Hermitian cones among the surfaces of degree $q+1$ of $\text{PG}(3,q^{2})$ by their intersection numbers with planes. We then use this result and provide a characterization of nonsingular Hermitian varieties of $\text{PG}(4,q^{2})$ among quasi-Hermitian ones.

We revisit the coordinatisation method for projective planes by considering the consequences of using finite fields to coordinatise projective planes of prime power order. This leads to some general restrictions on the form of the resulting planar ternary ring (PTR) when viewed as a trivariate polynomial over the field. We also consider how the Lenz–Barlotti type of the plane being coordinatised impacts the form of the PTR polynomial, thereby deriving further restrictions.

We prove the existence and give constructions of a $(p(k)-1)$-fold perfect resolvable $(v,k,1)$-Mendelsohn design for any integers $v>k\geq 2$ with $v\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}\,k$ such that there exists a finite Frobenius group whose kernel $K$ has order $v$ and whose complement contains an element $\unicode[STIX]{x1D719}$ of order $k$, where $p(k)$ is the least prime factor of $k$. Such a design admits $K\rtimes \langle \unicode[STIX]{x1D719}\rangle$ as a group of automorphisms and is perfect when $k$ is a prime. As an application we prove that for any integer $v=p_{1}^{e_{1}}\cdots p_{t}^{e_{t}}\geq 3$ in prime factorisation and any prime $k$ dividing $p_{i}^{e_{i}}-1$ for $1\leq i\leq t$, there exists a resolvable perfect $(v,k,1)$-Mendelsohn design that admits a Frobenius group as a group of automorphisms. We also prove that, if $k$ is even and divides $p_{i}-1$ for $1\leq i\leq t$, then there are at least $\unicode[STIX]{x1D711}(k)^{t}$ resolvable $(v,k,1)$-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where $\unicode[STIX]{x1D711}$ is Euler’s totient function.

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame, which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that certain decompositions of tuples of coordinate vector fields are related by certain elementary transformations. This is shown with a nontrivial result on matroid partition.

We solve a problem posed by Cardinali and Sastry (Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order. Proc. Indian Acad. Sci. Math. Sci.126 (2016), 591–612) about factorization of 2-covers of finite classical generalized quadrangles (GQs). To that end, we develop a general theory of cover factorization for GQs, and in particular, we study the isomorphism problem for such covers and associated geometries. As a byproduct, we obtain new results about semi-partial geometries coming from θ-covers, and consider related problems.

Let $A$ be a set of natural numbers. Recent work has suggested a strong link between the additive energy of $A$ (the number of solutions to $a_{1}+a_{2}=a_{3}+a_{4}$ with $a_{i}\in A$) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of $A$ modulo $1$. There appears to be reasonable evidence to speculate a sharp Khinchin-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian.

It is proved that every non-trivial Latin square has an upper embedding in a non-orientable surface and every Latin square of odd order has an upper embedding in an orientable surface. In the latter case, detailed results about the possible automorphisms and their actions are also obtained.

This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős–Szekeres theorem: For every k ≥ 1, every order-nk-dimensional permutation contains a monotone subsequence of length Ωk($\sqrt{n}$), and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random k-dimensional permutation of order n is asymptotically almost surely Θk(nk/(k+1)).

The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on $G$, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that $G$ cannot have holomorph compound O’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.

A collection of $k$ sets is said to form a $k$-sunflower, or $\unicode[STIX]{x1D6E5}$-system, if the intersection of any two sets from the collection is the same, and we call a family of sets ${\mathcal{F}}$sunflower-free if it contains no $3$-sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt (‘On large subsets of $\mathbb{F}_{q}^{n}$ with no three-term arithmetic progression’, Ann. of Math. (2) 185 (2017), 339–343); (‘Progression-free sets in $\mathbb{Z}_{4}^{n}$ are exponentially small’, Ann. of Math. (2) 185 (2017), 331–337) we apply the polynomial method directly to Erdős–Szemerédi sunflower problem (Erdős and Szemerédi, ‘Combinatorial properties of systems of sets’, J. Combin. Theory Ser. A 24 (1978), 308–313) and prove that any sunflower-free family ${\mathcal{F}}$ of subsets of $\{1,2,\ldots ,n\}$ has size at most

$$\begin{eqnarray}|{\mathcal{F}}|\leqslant 3n\mathop{\sum }_{k\leqslant n/3}\binom{n}{k}\leqslant \left(\frac{3}{2^{2/3}}\right)^{n(1+o(1))}.\end{eqnarray}$$

$A\subset (\mathbb{Z}/D\mathbb{Z})^{n}=\{1,2,\ldots ,D\}^{n}$

$D>2$

$x,y,z\in A$

$i$

$x_{i},y_{i},z_{i}$

$\unicode[STIX]{x1D712}:\mathbb{Z}/D\mathbb{Z}\rightarrow \mathbb{C}$

$A\subset (\mathbb{Z}/D\mathbb{Z})^{n}$

$$\begin{eqnarray}|A|\leqslant c_{D}^{n}\end{eqnarray}$$

$c_{D}=\frac{3}{2^{2/3}}(D-1)^{2/3}$

$c_{D}\leqslant C$

$C$

$D$

A Latin square is reduced if its first row and first column are in natural order. For Latin squares of a particular order $n$, there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order $n\rightarrow \infty$.

In the full rectangular version of Gilbert's planar tessellation (see Gilbert (1967), Mackisack and Miles (1996), and Burridge et al. (2013)), lines extend either horizontally (with east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a stationary Poisson point process, each ray stopping when it meets another ray that has blocked its path. In the half-Gilbert rectangular version, east- and south-growing rays, whilst having the potential to block each other, do not interact with west and north rays, and vice versa. East- and south-growing rays have a reciprocality of blocking, as do west and north. In this paper we significantly expand and simplify the half-Gilbert analytic results that we gave in Burridge et al. (2013). We also show how the idea of reciprocality of blocking between rays can be used in a much wider context, with rays not necessarily orthogonal and with seeds that produce more than two rays.

In this paper, we consider the so-called “Furstenberg set problem” in high dimensions. First, following Wolff’s work on the two-dimensional real case, we provide “reasonable” upper bounds for the problem for $\mathbb{R}$ or $\mathbb{F}_{p}$. Next we study the “critical” case and improve the “trivial” exponent by ${\rm\Omega}(1/n^{2})$ for $\mathbb{F}_{p}^{n}$. Our key tool in obtaining this lower bound is a theorem about how things behave when the Loomis–Whitney inequality is nearly sharp, as it helps us to reduce the problem to dimension two.

We show that if a Barker sequence of length $n>13$ exists, then either n $=$ 3 979 201 339 721 749 133 016 171 583 224 100, or $n > 4\cdot 10^{33}$. This improves the lower bound on the length of a long Barker sequence by a factor of nearly $2000$. We also obtain eighteen additional integers $n<10^{50}$ that cannot be ruled out as the length of a Barker sequence, and find more than 237 000 additional candidates $n<10^{100}$. These results are obtained by completing extensive searches for Wieferich prime pairs and using them, together with a number of arithmetic restrictions on $n$, to construct qualifying integers below a given bound. We also report on some updated computations regarding open cases of the circulant Hadamard matrix problem.

While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown–Goodearl–Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch–Kresch–Tamvakis’ approaches to quantum Schubert calculus.