We attempt to develop a new chapter of the theory of uniform distribution; we call it strong uniformity. Strong uniformity in a nutshell means that we combine Lebesgue measure with the classical theory of uniform distribution, basically founded by Weyl in his famous paper from 1916 [Über die Gleichverteilung von Zahlen mod Eins, Math. Ann.77 (1916), 313–352], which is built around nice test sets, such as axis-parallel rectangles and boxes. We prove the continuous version of the well-known Khinchin’s conjecture [Eins Satz über Kettenbrüche mit arithmetischen Adwendungen, Math. Z.18 (1923), 289–306] in every dimension $d\geqslant 2$ (the discrete version turned out to be false—it was disproved by Marstrand [On Khinchin’s conjecture about strong uniform distribution, Proc. Lond. Math. Soc. (3) 21 (1970), 540–556]). We consider an arbitrarily complicated but fixed measurable test set $S$ in the $d$-dimensional unit cube, and study the uniformity of a typical member of some natural families of curves, such as all torus lines or billiard paths starting from the origin, with respect to $S$. In the two-dimensional case we have the very surprising superuniformity of the typical torus lines and billiard paths. In dimensions ${\geqslant}3$ we still have strong uniformity, but not superuniformity. However, in dimension three we have the even more striking super-duper uniformity for two-dimensional rays (replacing the torus lines). Finally, we indicate how to exhibit superuniform motions on every “reasonable” plane region (e.g., the circular disk) and on every “reasonable” closed surface (sphere, torus and so on).

We estimate the number of solutions to $|y^{2}-x^{3}|\leqslant X$ with $N\leqslant y\leqslant 2N$, in terms of both $N$ and $X$.

We show that the exponent of distribution of the ternary divisor function $d_{3}$ in arithmetic progressions to prime moduli is at least $1/2+1/46$, improving results of Friedlander–Iwaniec and Heath-Brown. Furthermore, when averaging over a fixed residue class, we prove that this exponent is increased to $1/2+1/34$.

We prove fractional Sobolev–Poincaré inequalities in unbounded John domains and we characterize fractional Hardy inequalities there.

In this paper we construct some Feller semigroups, hence Feller processes, with state space $\mathbb{R}^{n}\times \mathbb{Z}^{m}$ starting with pseudo-differential operators having symbols defined on $\mathbb{R}^{n}\times \mathbb{R}^{n}\times \mathbb{Z}^{m}\times \mathbb{T}^{m}$.

An integer $d$ is called a jumping champion for a given $x$ if $d$ is the most common gap between consecutive primes up to $x$. Occasionally, several gaps are equally common. Hence, there can be more than one jumping champion for the same $x$. In 1999, Odlyzko et al provided convincing heuristics and empirical evidence for the truth of the hypothesis that the jumping champions greater than 1 are 4 and the primorials $2,6,30,210,2310,\ldots \,$. In this paper, we prove that an appropriate form of the Hardy–Littlewood prime $k$-tuple conjecture for prime pairs and prime triples implies that all sufficiently large jumping champions are primorials and that all sufficiently large primorials are jumping champions over a long range of $x$.

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.

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalizations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces $H^{s}({\rm\Omega})$ and $\widetilde{H}^{s}({\rm\Omega})$, for $s\in \mathbb{R}$ and an open ${\rm\Omega}\subset \mathbb{R}^{n}$. We exhibit examples in one and two dimensions of sets ${\rm\Omega}$ for which these scales of Sobolev spaces are not interpolation scales. In the cases where they are interpolation scales (in particular, if ${\rm\Omega}$ is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large.

By establishing an improved level of distribution we study almost-primes of the form $f(p,n)$ where $f$ is an irreducible binary form over $\mathbb{Z}$.

We study the geometry of the space of measures of a compact ultrametric space $X$, endowed with the $L^{p}$ Wasserstein distance from optimal transportation. We show that the power $p$ of this distance makes this Wasserstein space affinely isometric to a convex subset of $\ell ^{1}$. As a consequence, it is connected by $1/p$-Hölder arcs, but any ${\it\alpha}$-Hölder arc with ${\it\alpha}>1/p$ must be constant. This result is obtained via a reformulation of the distance between two measures which is very specific to the case when $X$ is ultrametric; however, thanks to the Mendel–Naor ultrametric skeleton it has consequences even when $X$ is a general compact metric space. More precisely, we use it to estimate the size of Wasserstein spaces, measured by an analogue of Hausdorff dimension that is adapted to (some) infinite-dimensional spaces. The result we get generalizes greatly our previous estimate, which needed a strong rectifiability assumption. The proof of this estimate involves a structural theorem of independent interest: every ultrametric space contains large co-Lipschitz images of regular ultrametric spaces, i.e. spaces of the form $\{1,\dots ,k\}^{\mathbb{N}}$ with a natural ultrametric. We are also led to an example of independent interest: a space of positive lower Minkowski dimension, all of whose proper closed subsets have vanishing lower Minkowski dimension.

We study the metric entropy of the metric space ${\mathcal{B}}_{n}$ of all $n$-dimensional Banach spaces (the so-called Banach–Mazur compactum) equipped with the Banach–Mazur (multiplicative) “distance” $d$. We are interested either in estimates independent of the dimension or in asymptotic estimates when the dimension tends to $\infty$. For instance, we prove that, if $N({\mathcal{B}}_{n},d,1+{\it\varepsilon})$ is the smallest number of “balls” of “radius” $1+{\it\varepsilon}$ that cover ${\mathcal{B}}_{n}$, then for any ${\it\varepsilon}>0$ we have

$$\begin{eqnarray}0<\liminf _{n\rightarrow \infty }n^{-1}\log \log N({\mathcal{B}}_{n},d,1+{\it\varepsilon})\leqslant \limsup _{n\rightarrow \infty }n^{-1}\log \log N({\mathcal{B}}_{n},d,1+{\it\varepsilon})<\infty .\end{eqnarray}$$

$n$

$d_{N}$

$N\times N$

$N$

$N$

$N=1$

$n$

$N\times N$

This paper presents a model of a 1D–1D dynamic multi-structure, supporting propagation of a transition wave. It is used to explain the recent phenomenon of the collapse of the San Saba bridge. An analytical model is supplied with illustrative numerical simulations.

Asymptotic analysis of the Hele-Shaw flow with a small moving obstacle is performed. The method of solution utilizes the uniform asymptotic formulas for Green’s and Neumann functions recently obtained by V. Maz’ya and A. Movchan. The theoretical results of the paper are illustrated by numerical simulations.

The purpose of the paper is to introduce a novel “splitting” procedure which can be helpful in the derivation of explicit formulas for various Bellman functions. As an illustration, we study the action of the dyadic maximal operator on $L^{p}$. The associated Bellman function $\mathfrak{B}_{p}$, introduced by Nazarov and Treil, was found explicitly by Melas with the use of combinatorial properties of the maximal operator, and was later rediscovered by Slavin, Stokolos and Vasyunin with the use of the corresponding Monge–Ampère partial differential equation. Our new argument enables an alternative simple derivation of $\mathfrak{B}_{p}$.

Let ${\it\alpha}\in \mathbb{C}$ in the upper half-plane and let $I$ be an interval. We construct an analogue of Selberg’s majorant of the characteristic function of $I$ that vanishes at the point ${\it\alpha}$. The construction is based on the solution to an extremal problem with positivity and interpolation constraints. Moreover, the passage from the auxiliary extremal problem to the construction of Selberg’s function with vanishing is easily adapted to provide analogous “majorants with vanishing” for any Beurling–Selberg majorant.

We study the system of Maxwell equations for a periodic composite dielectric medium with components whose dielectric permittivities ${\it\epsilon}$ have a high degree of contrast between each other. We assume that the ratio between the permittivities of the components with low and high values of ${\it\epsilon}$ is of the order ${\it\eta}^{2}$, where ${\it\eta}>0$ is the period of the medium. We determine the asymptotic behaviour of the electromagnetic response of such a medium in the “homogenization limit”, as ${\it\eta}\rightarrow 0$, and derive the limit system of Maxwell equations in $\mathbb{R}^{3}$. Our results extend a number of conclusions of a paper by Zhikov [On gaps in the spectrum of some divergent elliptic operators with periodic coefficients. St. Petersburg Math. J.16(5) (2004), 719–773] to the case of the full system of Maxwell equations.

We study translation invariant, real-valued valuations on the class of convex polytopes in Euclidean space and discuss which continuity properties are sufficient for an extension of such valuations to all convex bodies. For this purpose, we introduce flag support measures of convex bodies via a local Steiner formula and derive some of the properties of these measures.