arXiv subjects

Wayne M Lawton

Publications and source records attributed to Wayne M Lawton.

4 recordsLinked to original sources

arXiv2026

Topological and Diophantine properties of lattice subset projections

Fix $1 \leq n < m, k = m-n.$ The Grassmannian $Gr(n,m)$ is a compact $kn$-dimensional manifold with a unique rotation invariant probability measure $σ_n.$ For $W \in Gr(n,m)$, $P_W : \mathbb R^m \mapsto W$ is orthogonal projection. A lattice subset $L \subset \mathbb Z^m \subset \mathbb R^m$ is called $k$-dense if it intersects $C(O) := \bigcup_{V \in O} V\backslash \{0\}$ for every nonempty open $O \subset Gr(k,m)$. We use Baire's category theorem [4] to prove that $L$ is $k$-dense iff $L_{n,lim} := \{W \in Gr(n,m) : 0 \mbox{ is a limit point of } P_W(L) \}$ is a $G_δ$ set. We use Khintchine-Groshev's theorem [5,13,20] to characterize Diophantine properties of $L_{n,lim}$ by lacunary properties of $L$ and construct $k$-dense $L$ with $σ_n(L_{n,lim}) = 0$ and with $σ_n(L_{n,lim}) = 1.$ We pose related questions about the construction of multidimensional crystalline measures and Fourier quasicrystals.

math.NT↗

arXiv2026

Joint distribution of leftmost digits in positional notation and Schanuels's conjecture

Assume that $n \geq 2$ and $B = (b_1,...,b_n)$ has distince integer entries $\geq 3.$ For $x > 0$ let $d_B(x) := (d_{b_1}(x),...,d_{b_n}(x))$ where $d_{b_i}(x) \in \{1,...,b_i-1\}$ is the leftmost digit in the base-$b_i$ positional notation representation of $x.$ We prove that if $d_B$ is surjective, then $\ln b_i$ and $\ln b_j$ are rationally independent whenever $i \neq j.$ We prove the converse for $n = 2,$ and for $n \geq 3$ if $\{\ln p : p \mbox{ prime} \}$ is algebraically independent, a condition implied by Schanuel's conjecture about transcendental numbers.

math.NT↗

arXiv2025

Fourier Quasicrystals on $\mathbb R^n$

This paper has three aims. First, for $n \geq 1$ we construct a family of real-rooted trigonometric polynomial maps $P : \mathbb C^n \mapsto \mathbb C^n$ whose divisors are Fourier Quasicrystals (FQ). For $n = 1$ these divisors include the first nontrivial FQ with positive integer coefficients constructed by Kurasov and Sarnak [47, and for $n > 1$ they overlap with Meyer's curved model sets [65] and two-dimensional [66] and multidimensional [67] crystalline measures. We prove that the divisors are FQ by directly computing their Fourier transforms using a formula derived in [50].. Second, we extend the relationship between real-rootedness and amoebas, derived for $n = 1$ by Alon, Cohen and Vinzant [1], to the case $n > 1.$ The extension uses results in [10] about homology of complements of amoebas of algebraic sets of codimension $> 1.$ Third, we prove that the divisors of all uniformly generic real-rooted $P$ are FQ. The proof uses the formula relating Grothendieck residues and Newton polytopes derived by Gelfond and Khovanskii [34]. Finally, we note that Olevskii and Ulanovskii [72] have proved that all FQ with positive integer weights are divisors of real-rooted trigonometric polynomials for $n = 1$ but that the situation for $n > 1$ remains unsolved.

math.AG↗

arXiv2021

Tutorial on Rational Rotation $C^*$--Algebras

The rotation algebra $\mathcal A_θ$ is the universal $C^*$--algebra generated by unitary operators $U, V$ satisfying the commutation relation $UV = ωV U$ where $ω= e^{2πi θ}.$ They are rational if $θ= p/q$ with $1 \leq p \leq q-1,$ othewise irrational. Operators in these algebras relate to the quantum Hall effect \cite{boca,rammal,simon}, kicked quantum systems \cite{lawton1, wang}, and the spectacular solution of the Ten Martini problem \cite{avila}. Brabanter \cite{brabanter} and Yin \cite{yin} classified rational rotation $C^*$--algebras up to $*$-isomorphism. Stacey \cite{stacey} constructed their automorphism groups. They used methods known to experts: cocycles, crossed products, Dixmier-Douady classes, ergodic actions, K--theory, and Morita equivalence. This expository paper defines $\mathcal A_{p/q}$ as a $C^*$--algebra generated by two operators on a Hilbert space and uses linear algebra, Fourier series and the Gelfand-Naimark-Segal construction \cite{gelfand} to prove its universality. It then represents it as the algebra of sections of a matrix algebra bundle over a torus to compute its isomorphism class. The remarks section relates these concepts to general operator algebra theory. We write for mathematicians who are not $C^*$--algebra experts.

math.OA↗