Recent work by M. Afifurrahman established the first asymptotic estimates with error terms for the number of $2\times 2$ matrices with fixed non-zero determinant $n\in\mathbb{N}$, and with coefficients bounded in absolute value by X. In this paper we present a new proof of this result, which also gives an improved error term as $X\rightarrow\infty$. Similar to Afifurrahman’s result, our error term is uniform in both n and X, and our estimates are significant for X as small as $n^{1/2+\delta}$. To complement this, we also demonstrate that the exponent $1/2+\delta$ in this statement cannot be reduced, by establishing a result which gives a different asymptotic main term when n is either a prime or the square of a prime, and when $X=n^{1/2}$.

We establish a fixed-point theorem for the face maps that consist in deleting the ith entry of an ordered set. Furthermore, we show that there exists random finite sets of integers that are almost invariant under such deletions. Consequences for various monoids of order-preserving transformations of $\mathbf{N}$ are discussed in an appendix.

We show that any finitely presented group with an index two subgroup is realised as the fundamental group of a closed smooth non-orientable four-manifold that admits an exotic smooth structure, which is obtained by performing a Gluck twist. The orientation 2-covers of these four-manifolds are diffeomorphic. These two smooth structures remain inequivalent after adding arbitrarily many copies of the product of a pair of 2-spheres and stabilise after adding a single copy of the complex projective plane.

It is easy to see that every k-edge-colouring of the complete graph on $2^k+1$ vertices contains a monochromatic odd cycle. In 1973, Erdős and Graham asked to estimate the smallest L(k) such that every k-edge-colouring of $K_{2^k+1}$ contains a monochromatic odd cycle of length at most L(k). Recently, Girão and Hunter obtained the first nontrivial upper bound by showing that $L(k)=O({2^k}/({k^{1-o(1)}}))$, which improves the trivial bound by a polynomial factor. We obtain an exponential improvement by proving that $L(k)=O(k^{3/2}2^{k/2})$. Our proof combines tools from algebraic combinatorics and approximation theory.

In this paper, we investigate the extension of uniformisation results for Gromov hyperbolic spaces beyond the standard geodesic setting. By establishing a Gehring-Hayman type theorem for conformal deformations of any intrinsic Gromov hyperbolic space, we provide a framework for analysing spaces that do not necessarily admit geodesics. As a primary application, we prove that any complete intrinsic hyperbolic space with at least two points in the Gromov boundary can be uniformised by densities induced by Busemann functions. Furthermore, we establish that there exists a natural identification between the Gromov boundary of the original space and the metric boundary of the deformed space.

Gowers and Hatami initiated the inverse theory for the uniformity norms $U^k$ of matrix-valued functions on non-abelian groups by proving a 1%-inverse theorem for the $U^2$-norm and relating it to stability questions for almost representations. In this paper, we take a step toward an inverse theory for higher-order uniformity norms of matrix-valued functions on arbitrary groups by examining the 99% regime for the $U^k$-norm on perfect groups of bounded commutator width.

This analysis prompts a classification of Leibman’s quadratic maps between non-abelian groups. Our principal contribution is a complete description of these maps via an explicit universal construction. From this classification we deduce several applications: A full classification of quadratic maps on arbitrary abelian groups; a proof that no nontrivial polynomial maps of degree greater than one exist on perfect groups; stability results for approximate polynomial maps.

In 1972, Heilbronn introduced the notion of virtual characters and used it to study simple real zeros of Dedekind zeta functions. One of the consequences of his elegant work is the following. Let $\mathrm{K}/ \mathrm{F}$ be a Galois extension of number fields of odd degree. Then any real simple zero of $\zeta_{\mathrm{K}}$ is necessarily a simple zero of $\zeta_{\mathrm{F}}$. The ethos of this paper is to carry out investigation for arbitrary odd order real zeros. While the Riemann zeta function is conjectured to have only simple zeros, the same does not hold for arbitrary Dedekind zeta functions. One of the consequences of our work is that any Galois number field K of odd degree cannot have a non-trivial odd order real zero. Such parity is at least in conformity with extended Riemann hypothesis as the order of vanishing of the Dedekind zeta function $\zeta_{\mathrm{K}}$ at $1/2$ is necessarily even. We also indicate, via a number of illustrative examples (see Remarks 1·1 and 1·2), that in some sense our results are optimal.

In this paper we study the relationship between ideals and congruences of the tropical polynomial and Laurent polynomial semirings. We show that the variety of a non-zero prime ideal of the tropical (Laurent) polynomial semiring consists of at most one point. We also prove a result relating the dimension of an affine tropical variety and the dimension of its “coordinate semiring”.

Let $N \ge 1$, $k \ge 2$ even, and $\sigma$ denote a sign pattern for N. In this paper, we first determine the exact proportion of forms in $S_k(N)$ and $S_k^{\mathrm{new}}(N)$ with a given Atkin–Lehner sign pattern $\sigma$. Then we study the asymptotic behaviour of the Hecke operators $T_p$ over the subspaces of $S_k(N)$ and $S_k^{{\mathrm{new}}}(N)$ with Atkin–Lehner sign pattern $\sigma$. In particular, for the p-adic Plancherel measure $\mu_p$, we show that the Hecke eigenvalues for $T_p$ over these subspaces are $\mu_p$-equidistributed as $N+k \to \infty$.

We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients are not too large and do not differ from each other by too much. Additionally, we prove a number of multiplicity one type results for the number-theoretic objects attached to L-functions. These results follow from our main result, which has slightly weaker hypotheses than previous multiplicity one theorems for L-functions. Significantly stronger results are available when the L-function is known to be automorphic.