Recently, considerable attention has been given to the study of the arithmetic sum of two planar sets. We focus on understanding the interior (A + Γ)°, when Γ is a piecewise ${\mathcal C}^2$ curve and A ⊂ ℝ2. To begin, we give an example of a very large (full-measure, dense, Gδ) set A such that (A + S1)° = ∅, where S1 denotes the unit circle. This suggests that merely the size of A does not guarantee that (A + S1)° ≠ ∅. If, however, we assume that A is a kind of generalised product of two reasonably large sets, then (A + Γ)° ≠ ∅ whenever Γ has non-vanishing curvature. As a byproduct of our method, we prove that the pinned distance set of C := Cγ × Cγ, γ ⩾ 1/3, pinned at any point of C has non-empty interior, where Cγ (see (1.1)) is the middle 1 − 2γ Cantor set (including the usual middle-third Cantor set, C1/3). Our proof for the middle-third Cantor set requires a separate method. We also prove that C + S1 has non-empty interior.

We prove that every flat tensor-idempotent in the module category Mod- of a tensor-triangulated category comes from a unique smashing ideal in . We deduce that the lattice of smashing ideals forms a frame.

We prove an upper bound for the fifth moment of Hecke L-functions associated to holomorphic Hecke cusp forms of full level and weight k in a dyadic interval K ≤ k ≤2K, as K → ∞. The bound is sharp on Selberg’s eigenvalue conjecture.

In recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to both the Fourier transform of classical Dedekind sums and the error term in the asymptotic formula for the sum of φ(n).

We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.

We analyse the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H.

In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained.

For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a superpolynomial upper bound.

Triple homomorphisms on C*-algebras and JB*-triples have been studied in the literature. From the viewpoint of associative algebras, we characterise the structure of triple homomorphisms from an arbitrary ⋆-algebra onto a prime *-algebra. As an application, we prove that every triple homomorphism from a Banach ⋆-algebra onto a prime semisimple idempotent Banach *-algebra is continuous. The analogous results for prime C*-algebras and standard operator *-algebras on Hilbert spaces are also described.

In this paper, we investigate the weak forms of the 2-part of the conjecture of Birch and Swinnerton-Dyer, and prove a lower bound for the 2-adic valuation of the algebraic part of the central value of the complex L-series for the family of quadratic twists of all optimal elliptic curves over ${\mathbb Q}$.

A generalisation of von Staudt’s theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the projective semilinear group over an algebraically closed field of transcendence degree at least 1 is 4-transitive.

It is well known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group G can be detected on the cohomology group H1(G,R[G]), where R is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group G the same information about the number of ends of G in the sense of H. Abels can be provided by dH1(G, Bi(G)), where Bi(G) is the rational discrete standard bimodule of G, and dH•(G, _) denotes rational discrete cohomology as introduced in [6].

As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B).

It is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.

A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later, in 1921, he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this paper we explain these flaws and prove that every cyclic group of composite order is a B-group, using only Burnside’s character-theoretic methods. We also survey the related literature, prove some new results on B-groups of prime-power order, state two related open problems and present some new computational data.

We prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A ⊆ G is k-stable. Then there is a normal subgroup H ≤ G of index at most n, and a set Y ⊆ G, which is a union of cosets of H, such that |A △ Y| ≤ε|H|. It follows that, for any coset C of H, either |C ∩ A|≤ ε|H| or |C \ A| ≤ ε |H|. This qualitatively generalises recent work of Terry and Wolf on vector spaces over $\mathbb{F}_p$.