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 $d\geq 2$ .

We examine a semigroup analogue of the Kumjian–Renault representation of C*-algebras with Cartan subalgebras on twisted groupoids. Specifically, we represent semigroups with distinguished normal subsemigroups as ‘slice-sections’ of groupoid bundles.

We consider the two-dimensional shrinking target problem in beta dynamical systems (for general $\beta>1$ ) with general errors of approximation. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in [0,1]$ , define the shrinking target set

$$ \begin{align*}E(T_\beta, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{@{}ll@{}} \lvert T_{\beta}^{n}x-x_{0}\rvert <e^{-S_nf(x)}\\[1ex] \lvert T_{\beta}^{n}y-y_{0}\rvert < e^{-S_ng(y)} \end{array} \ {\text{for infinitely many}} \ n\in \mathbb N \right\}, \end{align*} $$

where $S_nf(x)=\sum _{j=0}^{n-1}f(T_\beta ^jx)$ is the Birkhoff sum. We calculate the Hausdorff dimension of this set and prove that it is the solution to some pressure function. This represents the first result of this kind for the higher-dimensional beta dynamical systems.

In this paper we study the existence of higher dimensional arithmetic progressions in Meyer sets. We show that the case when the ratios are linearly dependent over ${\mathbb Z}$ is trivial and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set $\Lambda $ and a fully Euclidean model set with the property that finitely many translates of cover $\Lambda $ , we prove that we can find higher dimensional arithmetic progressions of arbitrary length with k linearly independent ratios in $\Lambda $ if and only if k is at most the rank of the ${\mathbb Z}$ -module generated by . We use this result to characterize the Meyer sets that are subsets of fully Euclidean model sets.

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $\Delta (\lambda )$ to be such that every nonzero homomorphism from another Verma supermodule to $\Delta (\lambda )$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $\mathfrak {pe} (n)$ and, furthermore, to reduce the problem of description of $\mathrm {Ext}^1_{\mathcal O}(L(\mu ),\Delta (\lambda ))$ for $\mathfrak {pe} (n)$ to the similar problem for the Lie algebra $\mathfrak {gl}(n)$ . Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category $\mathcal O^{\mathfrak {p}}$ for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra $\mathfrak {pe} (n)$ and the orthosymplectic Lie superalgebra $\mathfrak {osp}(2|2n)$ .

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 $(-2, 2)$ .

The ring $\mathbb Z_{d}$ of d-adic integers has a natural interpretation as the boundary of a rooted d-ary tree $T_{d}$ . Endomorphisms of this tree (that is, solenoidal maps) are in one-to-one correspondence with 1-Lipschitz mappings from $\mathbb Z_{d}$ to itself. In the case when $d=p$ is prime, Anashin [‘Automata finiteness criterion in terms of van der Put series of automata functions’,p-Adic Numbers Ultrametric Anal. Appl. 4(2) (2012), 151–160] showed that $f\in \mathrm {Lip}^{1}(\mathbb Z_{p})$ is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute a p-automatic sequence over a finite subset of $\mathbb Z_{p}\cap \mathbb Q$ . We generalize this result to arbitrary integers $d\geq 2$ and describe the explicit connection between the Moore automaton producing such a sequence and the Mealy automaton inducing the corresponding endomorphism of a rooted tree. We also produce two algorithms converting one automaton to the other and vice versa. As a demonstration, we apply our algorithms to the Thue–Morse sequence and to one of the generators of the lamplighter group acting on the binary rooted tree.

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.

In this paper, we study the boundary blow-up problem related to the infinity Laplacian

$$ \begin{align*}\begin{cases} \Delta_{\infty}^h u=u^q &\mathrm{in}\; \Omega, \\ u=\infty &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$

where $\Delta _{\infty }^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator associated with the infinity Laplacian arising from the stochastic game named Tug-of-War. When $q>h>1$ , we establish the existence of the boundary blow-up viscosity solution. Moreover, when the domain satisfies some regular condition, we establish the asymptotic estimate of the blow-up solution near the boundary. As an application of the asymptotic estimate and the comparison principle, we obtain the uniqueness result of the large solution. We also give the nonexistence of the large solution for the case $q \leq h.$

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.

The Thue–Morse sequence is a prototypical automatic sequence found in diverse areas of mathematics, and in computer science. We study occurrences of factors w within this sequence, or more precisely, the sequence of gaps between consecutive occurrences. This gap sequence is morphic; we prove that it is not automatic as soon as the length of w is at least $2$ , thereby answering a question by J. Shallit in the affirmative. We give an explicit method to compute the discrepancy of the number of occurrences of the block $\mathtt {01}$ in the Thue–Morse sequence. We prove that the sequence of discrepancies is the sequence of output sums of a certain base- $2$ transducer.

Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal. 158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.

We give an upper bound for the minimum s with the property that every sufficiently large integer can be represented as the sum of s positive kth powers of integers, each of which is represented as the sum of three positive cubes for the cases $2\leq k\leq 4.$

Savin [‘ $\mathcal {C}^{1}$ regularity for infinity harmonic functions in two dimensions’, Arch. Ration. Mech. Anal. 3(176) (2005), 351–361] proved that every planar absolutely minimizing Lipschitz (AML) function is continuously differentiable whenever the ambient space is Euclidean. More recently, Peng et al. [‘Regularity of absolute minimizers for continuous convex Hamiltonians’, J. Differential Equations 274 (2021), 1115–1164] proved that this property remains true for planar AML functions for certain convex Hamiltonians, using some Euclidean techniques. Their result can be applied to AML functions defined in two-dimensional normed spaces with differentiable norm. In this work we develop a purely non-Euclidean technique to obtain the regularity of planar AML functions in two-dimensional normed spaces with differentiable norm.