"Bulletin of the Brazilian Mathematical Society, New Series"

Ghaani Farashahi, Arash

doi: 10.1007/s00574-018-00129-6pmid: N/A

This paper presents an operator theory approach for the abstract structure of Banach function modules over coset spaces of compact subgroups. Let G be a locally compact group and H be a compact subgroup of G. Let μ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu $$\end{document} be the normalized G-invariant measure over the homogeneous space G / H associated to the Weil’s formula and 1≤p<∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1\le p<\infty $$\end{document}. We then introduce the notion of convolution left-module action of L1(G/H,μ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^1(G/H,\mu )$$\end{document} on the Banach function spaces Lp(G/H,μ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p(G/H,\mu )$$\end{document}.

Abdullah Sharaf, Khadijah; Bensouf, Aymen; Chtioui, Hichem; Soumaré, Abdellahi

doi: 10.1007/s00574-021-00262-9pmid: N/A

We consider the problem of finding conformal metrics on the unit ball Bn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {B}}^n$$\end{document} of Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^n$$\end{document}, n≥3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\ge 3$$\end{document}, with zero scalar curvature and prescribed mean curvature on the boundary. We study the lack of compactness of the problem and we prove an existence result based on an index-counting formula.

Rosales, Leobardo

doi: 10.1007/s00574-021-00263-8pmid: N/A

We study n-dimensional area-minimizing currents T in Rn+1,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbf {R}^{n+1},$$\end{document} with boundary ∂T\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial T$$\end{document} satisfying two properties: ∂T\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial T$$\end{document} is locally a finite sum of (n-1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(n-1)$$\end{document}-dimensional C1,α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^{1,\alpha }$$\end{document} orientable submanifolds which only meet tangentially and with same orientation, for some α∈(0,1]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (0,1]$$\end{document}; ∂T\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial T$$\end{document} has mean curvature =hνT\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$=h \nu _{T}$$\end{document} where h is a Lipschitz scalar-valued function and νT\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nu _{T}$$\end{document} is the generalized outward pointing normal of ∂T\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial T$$\end{document} with respect to T. Similar to the proof of the Alexandrov reflection principle, we use a geometric maximum principle argument to give a partial boundary regularity result for such currents T. We show that near any point x in the support of ∂T,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial T,$$\end{document} either the support of T has very uncontrolled structure, or the support of T near x is the finite union of orientable C1,α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^{1,\alpha }$$\end{document} hypersurfaces-with-boundary with disjoint interiors and common boundary points only along the support of ∂T\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial T$$\end{document}.

Caetano, Vladimir; Hinojosa, Gabriela; Valdez, Rogelio

doi: 10.1007/s00574-021-00264-7pmid: N/A

Let T be a pearl necklace consisting of the union of n consecutive tangent closed 3-balls Bi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_i$$\end{document} (i=1,2,…,n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$i=1,2,\ldots , n$$\end{document}) and consider the Kleininan group ΓT\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _T$$\end{document} generated by the reflections on the boundaries ∂Bi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial B_i$$\end{document}. Let Λ(ΓT)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Lambda (\Gamma _T)$$\end{document} be a wild knot obtained as the limit set of ΓT\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _T$$\end{document} acting on the 3-sphere S3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {S}^3$$\end{document}. We say that a n-pearl necklace V consisting of the union of consecutive tangent closed 3-balls Ci\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C_i$$\end{document} (i=1,2,…,n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$i=1,2,\ldots , n$$\end{document}) is equivalent to T if there exists a homeomorphism φ:S3→S3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi :\mathbb {S}^3\rightarrow \mathbb {S}^3$$\end{document} such that φ(V)=T\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi (V)=T$$\end{document}, φ(Ci)=Bi\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi (C_i)=B_i$$\end{document}. and φ(Ci∩Ci+1)=Bi∩Bi+1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi (C_i\cap C_{i+1})=B_i\cap B_{i+1}$$\end{document}. In this paper, we prove that the Hausdorff dimension of Λ(ΓT)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Lambda (\Gamma _T)$$\end{document} varies continuously for equivalent pearl necklaces.

Menegon, Aurélio; Marques-Silva, Camila S.

doi: 10.1007/s00574-021-00265-6pmid: N/A

We describe the topology of the local Milnor fiber of a function f defined on a 3-dimensional subanalytic subset W⊂R3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$W \subset {\mathbb {R}}^3$$\end{document}, in terms of the embedded topological type of its link Kf\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_f$$\end{document}. Precisely, we prove that the interior of the local Milnor fiber of ‖f‖\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Vert f\Vert $$\end{document} is homeomorphic to the complement of Kf\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_f$$\end{document} in the corresponding sphere, extending a result due to Milnor to non-analytic situations. We also prove that the topology of the fiber does not change if we use a suitable fundamental system of neighborhoods instead of balls, extending to the real setting a classical result due to Lê and Teissier for complex functions.

Condori, Alexander; Carvalho, Silas L.

doi: 10.1007/s00574-021-00266-5pmid: N/A

We investigate in this work some situations where it is possible to estimate or determine the upper and the lower q-generalized fractal dimensions Dμ±(q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D^{\pm }_{\mu }(q)$$\end{document}, q∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q\in {\mathbb {R}}$$\end{document}, of invariant measures associated with continuous transformations over compact metric spaces. In particular, we present an alternative proof of Young’s Theorem by Young (Ergod. Theory Dyn. Syst. 2(1):109–124, 1982) for the generalized fractal dimensions of the Bowen-Margulis measure associated with a C1+α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^{1+\alpha }$$\end{document}-Axiom A system over a two-dimensional compact Riemannian manifold M. We also present estimates for the generalized fractal dimensions of an ergodic measure for which Brin-Katok’s Theorem is pointwise satisfied, in terms of its metric entropy. Furthermore, for expansive homeomorphisms (like C1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^1$$\end{document}-Axiom A systems), we show that the set of invariant measures such that Dμ+(q)=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D_\mu ^+(q)=0$$\end{document} (q≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q\ge 1$$\end{document}), under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each s∈[0,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s\in [0,1)$$\end{document}, Dμ+(s)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D^{+}_{\mu }(s)$$\end{document} is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric. Finally, we show that, for some dynamical systems, the metric entropy of an invariant measure is typically zero, settling a conjecture posed by Sigmund (Trans. Am. Math. Soc. 190:285–299, 1974) for Lipschitz transformations which satisfy the specification property.

Camargo, André Pierro de; Martin, Paulo Agozzini

doi: 10.1007/s00574-021-00267-4pmid: N/A

In this note we exhibit some large sets Θx⊂{1,2,…,⌊x⌋}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varTheta _x \subset \{1, 2, \ldots , \lfloor x \rfloor \}$$\end{document} such that the sum of the Möbius function over Θx\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varTheta _x$$\end{document} is small and independent of x. We show that the existence of some of these sets are intimately connected with the existence of the alternating series used by Tschebyschef and Sylvester to bound the prime counter function Π(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varPi (x)$$\end{document}.

Botelho, Geraldo; Wood, Raquel

doi: 10.1007/s00574-021-00268-3pmid: N/A

This paper develops a technique to represent linear functionals on spaces of homogeneous polynomials between Banach spaces. This technique applies to cases that are not covered by the classical polynomial Borel transform. We provide applications to represent linear functionals on spaces of approximable, compact, hyper-nuclear and hyper-σ(p)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma (p)$$\end{document}-nuclear polynomials. An unexpected difference between the polynomial and multilinear theories is disclosed.

Gueye, Alioune; Rihane, Salah Eddine; Togbé, Alain

doi: 10.1007/s00574-021-00269-2pmid: N/A

Let k≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k\ge 2$$\end{document}. A generalization of the well-known Fibonacci sequence is the k-Fibonacci sequence. For this sequence, the first k terms are 0,…,0,1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0,\ldots ,0,1$$\end{document} and each term afterwards is the sum of the preceding k terms. In this paper, we find all k-Fibonacci which are product of two Fermat numbers.

Belolipetsky, Mikhail; Lalín, Matilde; Murillo, Plinio G. P.; Thompson, Lola

doi: 10.1007/s00574-021-00270-9pmid: 35646107

It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic 3-dimensional orbifold defines cQ1/2+O(Q1/4)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$c Q^{1/2} + O(Q^{1/4})$$\end{document} square-rootable Salem numbers of degree 4 which are less than or equal to Q. This quantity can be compared to the total number of such Salem numbers, which is shown to be asymptotic to 43Q3/2+O(Q)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{4}{3}Q^{3/2}+O(Q)$$\end{document}. Assuming the gap conjecture of Marklof, we can extend these results to compact arithmetic 3-orbifolds. As an application, we obtain lower bounds for the strong exponential growth of mean multiplicities in the geodesic spectrum of non-compact even dimensional arithmetic orbifolds. Previously, such lower bounds had only been obtained in dimensions 2 and 3.