A connected, locally finite graph $\Gamma $ is a Cayley–Abels graph for a totally disconnected, locally compact group G if G acts vertex-transitively on $\Gamma $ with compact, open vertex stabilizers. Define the minimal degree of G as the minimal degree of a Cayley–Abels graph of G. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if $T_{d}$ denotes the d-regular tree, then the minimal degree of $\mathrm{Aut}(T_{d})$ is d for all $d\geq 2$ .

In this paper, we study lower-order terms of the one-level density of low-lying zeros of quadratic Hecke L-functions in the Gaussian field. Assuming the generalized Riemann hypothesis, our result is valid for even test functions whose Fourier transforms are supported in $(-2, 2)$ . Moreover, we apply the ratios conjecture of L-functions to derive these lower-order terms as well. Up to the first lower-order term, we show that our results are consistent with each other when the Fourier transforms of the test functions are supported in $(-2, 2)$ .

In this paper we study the small-scale equidistribution property of random waves whose coefficients are determined by an unfair coin. That is, the coefficients take value $+1$ with probability p and $-1$ with probability $1-p$ . Random waves whose coefficients are associated with a fair coin are known to equidistribute down to the wavelength scale. We obtain explicit requirements on the deviation from the fair ( $p=0.5$ ) coin to retain equidistribution.

Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists.

We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.