We provide primitive recursive bounds for the finite version of Gowers’ $c_{0}$ theorem for both the positive and the general case. We also provide multidimensional versions of these results.

Two players take it in turn to claim edges from a graph $G$. The first player (“Maker”) wins if at any point he has claimed $s$ edges at a vertex without the second player (“Breaker”) having claimed a single edge at that vertex. If, by the end of play, this does not occur we say that Breaker wins. Our main aim is to show that for every $s$ there is a graph $G$ in which Maker has a winning strategy.

We address a question raised by Anderson, Hayman and Pommerenke relating to a classical result on univalent functions $f$ in the unit disc due to Spencer, and involving the size of the set of ${\it\theta}\in [-{\it\pi},{\it\pi}]$ for which we have $\log |f(r\text{e}^{\text{i}{\it\theta}})|\neq o(\log (1/(1-r)))$ as $r\rightarrow 1.$ An answer is given in terms of a certain generalized capacity, and also in terms of Hausdorff measure. Further results regarding the radial growth of univalent functions are also established, and some examples are constructed which relate to the sharpness of these results.

Given a finite-dimensional Banach space $X$ and an Auerbach basis $\{(x_{k},x_{k}^{\ast }):1\leqslant k\leqslant n\}$ of $X$, it is proved that there exist $n+1$ linear combinations $z_{1},\ldots ,z_{n+1}$ of $x_{1},\ldots ,x_{n}$ with coordinates $0,\pm 1$, such that $\Vert z_{k}\Vert =1$, for $k=1$, $2,\ldots ,n+1$ and $\Vert z_{k}-z_{l}\Vert >1$, for $1\leqslant k<l\leqslant n+1$.

Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the convex body. Since these coefficients are bounded by the dimension but possibly smaller, our inequalities sharpen the original ones. Since they can often be computed efficiently, the improved bounds may also be used to obtain better bounds in approximation algorithms.

We give a short and elementary proof of an inverse Bernstein-type inequality found by S. Khrushchev for the derivative of a polynomial having all its zeros on the unit circle. The inequality is used to show that equally-spaced points solve a min–max–min problem for the logarithmic potential of such polynomials. Using techniques recently developed for polarization (Chebyshev-type) problems, we show that this optimality also holds for a large class of potentials, including the Riesz potentials $1/r^{s}$ with $s>0.$

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 $X$.

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 $x$.

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 $\mathbb{Z}$.