We show that under certain general conditions, short sums of ℓ-adic trace functions over finite fields follow a normal distribution asymptotically when the origin varies, generalising results of Erdős–Davenport, Mak–Zaharescu and Lamzouri. In particular, this applies to exponential sums arising from Fourier transforms such as Kloosterman sums or Birch sums, as we can deduce from the works of Katz. By approximating the moments of traces of random matrices in monodromy groups, a quantitative version can be given as in Lamzouri's article, exhibiting a different phenomenon than the averaging from the central limit theorem.

We are interested in characterising pairs S, T of F-linear subspaces in a field extension L/F such that the linear span ST of the set of products of elements of S and of elements of T has small dimension. Our central result is a linear analogue of Vosper's Theorem, which gives the structure of vector spaces S, T in a prime extension L of a finite field F for which

\begin{linenomath}$$ \dim_FST =\dim_F S+\dim_F T-1, $$\end{linenomath}

By means of a quaternion algebra over $\mathbb{F}$q(t), we construct an infinite series of torsion free, simply transitive, irreducible lattices in PGL2($\mathbb{F}$q((t))) × PGL2($\mathbb{F}$q((t))). The lattices depend on an odd prime power q = pr and a parameter τ ∈ $\mathbb{F}$q×, τ ≠ 1, and are the fundamental group of a square complex with just one vertex and universal covering Tq+1 × Tq+1, a product of trees with constant valency q + 1.

Our lattices give rise via non-archimedian uniformization to smooth projective surfaces of general type over $\mathbb{F}$q((t)) with ample canonical class, Chern numbers c12 = 2 c2, trivial Albanese variety and non-reduced Picard scheme. For q = 3, the Zariski–Euler characteristic attains its minimal value χ = 1: the surface is a non-classical fake quadric.

In this paper, the author (1) compare subnormal closures of finite sets in a free group F; (2) obtains the limit for the series of subnormal closures of a single element in F; (3) proves that the exponential growth rate (exp.g.r.) $\lim_{n\to \infty}\sqrt[n]{g_H(n)}$, where gH(n) is the growth function of a subgroup H with respect to a finite free basis of F, exists for any subgroup H of the free group F; (4) gives sharp estimates from below for the exp.g.r. of subnormal subgroups in free groups; and (5) finds the cogrowth of the subnormal closures of free generators.

In a paper from 2010, Budarina, Dickinson and Levesley studied the rational approximation properties of curves parametrised by polynomials with integral coefficients in Euclidean space of arbitrary dimension. Assuming the dimension is at least three and excluding the case of linear dependence of the polynomials together with P(X) ≡ 1 over the rational number field, we establish proper generalisations of their main result.

We extend a lower bound of Munshi on sums over divisors of a number n which are less than a fixed power of n from the squarefree case to the general case. In the process we prove a lower bound on the entropy of a geometric distribution with finite support, as well as a lower bound on the probability that a random variable is less than its mean given that it satisfies a natural condition related to its third cumulant.

Consider a product system over the positive cone of a quasi-lattice ordered group. We construct a Fell bundle over an associated groupoid so that the cross-sectional algebra of the bundle is isomorphic to the Nica–Toeplitz algebra of the product system. Under the additional hypothesis that the left actions in the product system are implemented by injective homomorphisms, we show that the cross-sectional algebra of the restriction of the bundle to a natural boundary subgroupoid coincides with the Cuntz–Nica–Pimsner algebra of the product system. We apply these results to improve on existing sufficient conditions for nuclearity of the Nica–Toeplitz algebra and the Cuntz–Nica–Pimsner algebra, and for the Cuntz–Nica–Pimsner algebra to coincide with its co-universal quotient.