The Zarankiewicz problem asks for an estimate on z(m, n; s, t), the largest number of 1’s in an m × n matrix with all entries 0 or 1 containing no s × t submatrix consisting entirely of 1’s. We show that a classical upper bound for z(m, n; s, t) due to Kővári, Sós and Turán is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method.

We prove that the set of all endpoints of the Julia set of $f(z)=\exp\!(z)-1$ which escape to infinity under iteration of f is not homeomorphic to the rational Hilbert space $\mathfrak E$ . As a corollary, we show that the set of all points $z\in \mathbb C$ whose orbits either escape to $\infty$ or attract to 0 is path-connected. We extend these results to many other functions in the exponential family.

In this paper we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measures of Iterated Function Systems equipped with a probability distribution. We recover a classical existence and uniqueness result under a contraction-on-average assumption, prove generalised moment bounds from which tail estimates can be deduced, consider the convergence of the empirical measure of an associated Markov chain, and prove in many cases the Lipschitz continuity of the stationary measure when the system is perturbed, with as a consequence a “linear response formula” at almost every parameter of the perturbation.

We improve the best known sum-product estimates over the reals. We prove that

\[\max(|A+A|,|A+A|)\geq |A|^{\frac{4}{3} + \frac{2}{1167} - o(1)}\,,\]

$A\subset \mathbb {R}$

Furthermore,

\[|AA+AA|\geq |A|^{\frac{127}{80} - o(1)}\,.\]

\[|A+A|\geq |A|^{\frac{30}{19}-o(1)}\,.\]

Given disjoint subsets T1, …, Tm of “not too large” primes up to x, we establish that for a random integer n drawn from [1, x], the m-dimensional vector enumerating the number of prime factors of n from T1, …, Tm converges to a vector of m independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when T1, …, Tm are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors.

A number of spectrum constructions have been devised to extract topological spaces from algebraic data. Prominent examples include the Zariski spectrum of a commutative ring, the Stone spectrum of a bounded distributive lattice, the Gelfand spectrum of a commutative unital C*-algebra and the Hofmann–Lawson spectrum of a continuous frame.

Inspired by the examples above, we define a spectrum for localic semirings. We use arguments in the symmetric monoidal category of suplattices to prove that, under conditions satisfied by the aforementioned examples, the spectrum can be constructed as the frame of overt weakly closed radical ideals and that it reduces to the usual constructions in those cases. Our proofs are constructive.

Our approach actually gives ‘quantalic’ spectrum from which the more familiar localic spectrum can then be derived. For a discrete ring this yields the quantale of ideals and in general should contain additional ‘differential’ information about the semiring.

Let A be an abelian variety defined over a number field k, let p be an odd prime number and let $F/k$ be a cyclic extension of p-power degree. Under not-too-stringent hypotheses we give an interpretation of the p-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of A of the ${\rm Gal}(F/k)$ -valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in situations in which the p-completion of the Mordell–Weil group of A over F is not a projective Galois module.

We prove uniform bounds for the Petersson norm of the cuspidal part of the theta series. This gives an improved asymptotic formula for the number of representations by a quadratic form. As an application, we show that every integer $n \neq 0,4,7 \,(\textrm{mod}\ 8)$ is represented as $n= x_1^2 + x_2^2 + x_3^3$ for integers $x_1,x_2,x_3$ such that the product $x_1x_2x_3$ has at most 72 prime divisors.

For a locally compact metrisable group G, we study the action of ${\rm Aut}(G)$ on ${\rm Sub}_G$ , the set of closed subgroups of G endowed with the Chabauty topology. Given an automorphism T of G, we relate the distality of the T-action on ${\rm Sub}_G$ with that of the T-action on G under a certain condition. If G is a connected Lie group, we characterise the distality of the T-action on ${\rm Sub}_G$ in terms of compactness of the closed subgroup generated by T in ${\rm Aut}(G)$ under certain conditions on the center of G or on T as follows: G has no compact central subgroup of positive dimension or T is unipotent or T is contained in the connected component of the identity in ${\rm Aut}(G)$ . Moreover, we also show that a connected Lie group G acts distally on ${\rm Sub}_G$ if and only if G is either compact or it is isomorphic to a direct product of a compact group and a vector group. All the results on the Lie groups mentioned above hold for the action on ${\rm Sub}^a_G$ , a subset of ${\rm Sub}_G$ consisting of closed abelian subgroups of G.

We prove that for positive integers $m \geq 1, n \geq 1$ and a prime number $p \neq 2,3$ there are finitely many finite m-generated Moufang loops of exponent $p^n$ .

Let $B^{H}$ be a fractional Brownian motion in $\mathbb{R}^{d}$ of Hurst index $H\in\left(0,1\right)$ , $f\;:\;\left[0,1\right]\longrightarrow\mathbb{R}^{d}$ a Borel function and $A\subset\left[0,1\right]$ a Borel set. We provide sufficient conditions for the image $(B^{H}+f)(A)$ to have a positive Lebesgue measure or to have a non-empty interior. This is done through the study of the properties of the density of the occupation measure of $(B^{H}+f)$ . Precisely, we prove that if the parabolic Hausdorff dimension of the graph of f is greater than Hd, then the density is a square integrable function. If, on the other hand, the Hausdorff dimension of A is greater than Hd, then it even admits a continuous version. This allows us to establish the result already cited.

Among the knots that are the connected sum of two torus knots with cobordism distance 1, we characterise those that have 4-dimensional clasp number at least 2, and we show that their n-fold connected self-sum has 4-dimensional clasp number at least 2n. Our proof works in the topological category. To contrast this, we build a family of topologically slice knots for which the n-fold connected self-sum has 4-ball genus n and 4-dimensional clasp number at least 2n.

We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers.

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings’s methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.