We discuss the problem of deciding when a metrisable topological group G has a canonically defined local Lipschitz geometry. This naturally leads to the concept of minimal metrics on G, that we characterise intrinsically in terms of a linear growth condition on powers of group elements. Combining this with work on the large scale geometry of topological groups, we also identify the class of metrisable groups admitting a canonical global Lipschitz geometry. In turn, minimal metrics connect with Hilbert’s fifth problem for completely metrisable groups and we show, assuming that the set of squares is sufficiently rich, that every element of some identity neighbourhood belongs to a 1-parameter subgroup.

We study the multiplicative Hilbert matrix, i.e. the infinite matrix with entries (mn−−−√log(mn))−1 for m,n≥2. This matrix was recently introduced within the context of the theory of Dirichlet series, and it was shown that the multiplicative Hilbert matrix has no eigenvalues and that its continuous spectrum coincides with [0,π]. Here we prove that the multiplicative Hilbert matrix has no singular continuous spectrum and that its absolutely continuous spectrum has multiplicity one. Our argument relies on spectral perturbation theory and scattering theory. Finding an explicit diagonalisation of the multiplicative Hilbert matrix remains an interesting open problem.

It is shown for the non-centered Hardy–Littlewood maximal operator M that ∥DMf∥1≤Cn∥Df∥1 for all radial functions in W1,1(Rn).

We aim to maximize the number of first-quadrant lattice points in a convex domain with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the stretch factor approaches 1 as the “radius” approaches infinity. In particular, the result implies that among all p-ellipses (or Lamé curves), the p-circle encloses the most first-quadrant lattice points as the radius approaches infinity, for 1<p<∞. The case p=2 corresponds to minimization of high eigenvalues of the Dirichlet Laplacian on rectangles, and so our work generalizes a result of Antunes and Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of van den Berg, Bucur and Gittins. Further, Ariturk and Laugesen recently handled 0<p<1 by building on our results here. The case p=1 remains open, and is closely related to minimizing energy levels of harmonic oscillators: which right triangles in the first quadrant with two sides along the axes will enclose the most lattice points, as the area tends to infinity?

For families of continuous plurisubharmonic functions we show that, in a local sense, separately bounded above implies bounded above.