In two dimensions, Gallagher’s theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fiber version of Gallagher’s theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes, and Velani. The idea is to find large generalized arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin–Schaeffer theorem for the problem at hand, via the geometry of numbers.

We prove that the fundamental group of any integer homology $3$-sphere different from the $3$-sphere admits irreducible representations of its fundamental group in $SL(2,\mathbb{C})$. For hyperbolic integer homology spheres, this comes with the definition; for Seifert-fibered integer homology spheres, this is well known. We prove that the splicing of any two nontrivial knots in $S^3$ admits an irreducible $SU\left(2\right)$-representation. Using a result of Kuperberg, we get the corollary that the problem of $3$-sphere recognition is in the complexity class $\mathsf{coNP}$, provided the generalized Riemann hypothesis holds. To prove our result, we establish a topological fact about the image of the $SU\left(2\right)$-representation variety of a nontrivial knot complement into the representation variety of its boundary torus, a pillowcase, using holonomy perturbations of the Chern–Simons function in an exhaustive way—showing that any area-preserving self-map of the pillowcase fixing the four singular points, and which is isotopic to the identity, can be $C^0$-approximated by maps realized geometrically through holonomy perturbations of the flatness equation in a thickened torus. We conclude with a stretching argument in instanton gauge theory and a nonvanishing result of Kronheimer and Mrowka for Donaldson’s invariants of a $4$-manifold which contains the $0$-surgery of a knot as a splitting hypersurface.

Let $\Gamma$ be either the automorphism group of the free group of rank $n\ge 4$ or the mapping class group of an orientable surface of genus $n\ge 12$ with at most $1$ boundary component, and let $G$ be either the subgroup of $\mathrm{IA}$-automorphisms or the Torelli subgroup of $\Gamma$. For $N\in \mathbb{N}$ denote by ${\gamma }_{N}G$ the $N$th term of the lower central series of $G$. We prove that

(i) any subgroup of $G$ containing ${\gamma }_{2}G=[G,G]$ (in particular, the Johnson kernel in the mapping class group case) is finitely generated;

(ii) if $N=2$ or $n\ge 8N-4$ and $K$ is any subgroup of $G$ containing ${\gamma }_{N}G$ (for instance, $K$ can be the $N$th term of the Johnson filtration of $G$), then $G/[K,K]$ is nilpotent and hence the Abelianization of $K$ is finitely generated;

(iii) if $H$ is any finite-index subgroup of $\Gamma$ containing ${\gamma }_{N}G$, with $N$ as in (ii), then $H$ has finite Abelianization.

We consider the Schrödinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time, and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data and remain when an extra power-like nonlinearity is present in the equation. One of the key steps of the proof consists in working in hydrodynamical variables to reduce the equation to a variant of the isothermal compressible Euler equation, whose large time behavior turns out to be governed by a parabolic equation involving a Fokker–Planck operator.