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To each of the symmetry groups of the Platonic solids we adjoin a carefully designed involution yielding topological generators of PU(2) which have optimal covering properties as well as efficient navigation. These are a consequence of optimal strong approximation for integral quadratic forms associated with certain special quaternion algebras and their arithmetic groups. The generators give super efficient 1-qubit quantum gates and are natural building blocks for the design of universal quantum gates.
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In a seminal series of papers from the 80s, Lubotzky, Phillips and Sarnak applied the Ramanujan-Petersson Conjecture for $GL_{2}$ (Delignes theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat-Tits trees associated with $SL_{2}(mathbb{Q}_{p})$. As a result, they obtained explicit Ramanujan Cayley graphs from $PSL_{2}left(mathbb{F}_{p}right)$, as well as optimal topological generators (Golden Gates) for the compact Lie group $PU(2)$. In higher dimension, the naive generalization of the Ramanujan Conjecture fails, due to the phenomenon of endoscopic lifts. In this paper we overcome this problem for $PU_{3}$ by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat-Tits buildings associated with $SL_{3}(mathbb{Q}_{p})$ and $SU_{3}(mathbb{Q}_{p})$, while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from $PSL_{3}(mathbb{F}_{p})$ and $PU_{3}left(mathbb{F}_{p}right)$, as well as golden gates for $PU(3)$.
Let $G$ be a connected, absolutely almost simple, algebraic group defined over a finitely generated, infinite field $K$, and let $Gamma$ be a Zariski dense subgroup of $G(K)$. We show, apart from some few exceptions, that the commensurability class of the field $mathcal{F}$ given by the compositum of the splitting fields of characteristic polynomials of generic elements of $Gamma$ determines the group $G$ upto isogeny over the algebraic closure of $K$.
We establish an analogue of the classical Polya-Vinogradov inequality for $GL(2, F_p)$, where $p$ is a prime. In the process, we compute the `singular Gauss sums for $GL(2, F_p)$. As an application, we show that the collection of elements in $GL(2,Z)$ whose reduction modulo $p$ are of maximal order in $GL(2, F_p)$ and whose matrix entries are bounded by $x$ has the expected size as soon as $xgg p^{1/2+ep}$ for any $ep>0$. In particular, there exist elements in $GL(2,Z)$ with matrix entries that are of the order $O(p^{1/2+ep})$ whose reduction modulo $p$ are primitive elements.
For a hyperbolic surface embedded eigenvalues of the Laplace operator are unstable and tend to become resonances. A sufficient dissolving condition was identified by Phillips-Sarnak and is elegantly expressed in Fermis Golden Rule. We prove formulas for higher approximations and obtain necessary and sufficient conditions for dissolving a cusp form with eigenfunction $u_j$ into a resonance. In the framework of perturbations in character varieties, we relate the result to the special values of the $L$-series $L(u_jotimes F^n, s)$. This is the Rankin-Selberg convolution of $u_j$ with $F(z)^n$, where $F(z)$ is the antiderivative of a weight 2 cusp form. In an example we show that the above-mentioned conditions force the embedded eigenvalue to become a resonance in a punctured neighborhood of the deformation space.
We show that an approximate lattice in a nilpotent Lie group admits a relatively dense subset of central $(1-epsilon)$-Bragg peaks for every $epsilon > 0$. For the Heisenberg group we deduce that the union of horizontal and vertical $(1-epsilon)$-Bragg peaks is relatively dense in the unitary dual. More generally we study uniform approximate lattices in extensions of lcsc groups. We obtain necesary and sufficient conditions for the existence of a continuous horizontal factor of the associated hull-dynamical system, and study the spectral theory of the hull-dynamical system relative to this horizontal factor.
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