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Florin P. Boca

Publications and source records attributed to Florin P. Boca.

At least 19 recordsLinked to original sources
arXiv2026

On denominators of consecutive $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions

The sequence $({\mathscr S}_Q)_Q$ of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions was defined in our previous work by ${\mathscr S}_Q := \{ a/q \in {\mathbb Q} \cap (0,1]: q+a+\bar{a} \le Q\}$, where $\bar{a}$ is the multiplicative inverse of $a\pmod{q}$ in $[1,q)$. Here, we prove that the set of $Q$-scaled denominators of consecutive fractions in ${\mathscr S}_Q$ is dense in the region ${\mathcal V}:=\{ (x,y)\in [0,1]^2 : \max \{ (1-3x)/2,2x-1\} \le y \le \max \{ x,1-x\} \}$, and provide a formula for their distribution in ${\mathcal V}$ as $Q\rightarrow \infty$.

math.NT↗
arXiv2025

On the distribution of $\operatorname{SL}(2,{\mathbb N})$-saturated Farey fractions

We consider the set ${\mathscr S}_Q$ of Farey fractions $d/b$ of order $Q$ with the property that there exists a matrix $\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right) \in \operatorname{SL}(2,{\mathbb Z})$ of trace at most $Q$, with positive entries and $a\ge \max\{ b,c\}$. For every $Q\ge 3$, the set ${\mathscr S}_Q \cup \{ 0\}$ is shown to define a unimodular partition of the interval $[0,1]$. We also prove that the elements of ${\mathscr S}_Q$ are asymptotically distributed with respect to the probability measure with density $(1/(1+x) -1/(2+x) )/\log (4/3) $ and that the sequence of sets $({\mathscr S}_Q)_Q$ has a limiting gap distribution as $Q\rightarrow \infty$.

math.NT↗
arXiv2025

Angular distribution towards the points of the neighbor-flips modular curve seen by a fast moving observer

Let $h$ be a fixed non-zero integer. For every $t\in \mathbb{R}_+$ and every prime $p$, consider the angles between rays from an observer located at the point $(-tJ_p^2,0)$ on the real axis towards the set of all integral solutions $(x,y)$ of the equation $y^{-1}-x^{-1}\equiv h \pmod{p}$ in the square $[-J_p,J_p]^2$, where $J_p=(p-1)/2$. We prove the existence of the limiting gap distribution for this set of angles as $p\rightarrow \infty$, providing explicit formulas for the corresponding density function, which turns out to be independent of $h$.

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arXiv2024

On the Gauss-Kuzmin-Lévy problem for nearest integer continued fractions

This note provides an effective bound in the Gauss-Kuzmin-Lévy problem for some Gauss type shifts associated with nearest integer continued fractions, acting on the interval $I_0=[0,\frac{1}{2}]$ or $I_0=[-\frac{1}{2},\frac{1}{2}]$. We prove asymptotic formulas $λ(T^{-n}I) =μ(I)(\vert I_0 \vert +O(q^n))$ for such transformations $T$, where $λ$ is the Lebesgue measure on $\mathbb R$, $μ$ the normalized $T$-invariant Lebesgue absolutely continuous measure, $I$ subinterval in $I_0$, and $q=0.288$ is smaller than the Wirsing constant $q_W=0.3036\ldots$

math.NT↗
arXiv2023

Distribution of angles to lattice points seen from a fast moving observer

We consider a square expanding with constant speed seen from an observer moving away with constant acceleration and study the distribution of angles between rays from the observer towards the lattice points in the square. We prove the existence of the gap distribution as time tends to infinity and provide explicit formulas for the corresponding density function.

math.NT↗
arXiv2021

Distribution of periodic points of certain Gauss shifts with infinite invariant measure

This paper investigates the periodic points of the Gauss type shifts associated to the even continued fraction (Schweiger) and to the backward continued fraction (Rényi). We show that they coincide exactly with two sets of quadratic irrationals that we call $E$-reduced, and respectively $B$-reduced. We prove that these numbers are equidistributed with respect to the (infinite) Lebesgue absolutely continuous invariant measures of the corresponding Gauss shift.

math.DS↗
arXiv2020

Products of matrices $[ \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}]$ and $[ \begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix} ]$ and the distribution of reduced quadratic irrationals

Let $Φ(N)$ denote the number of products of matrices $[ \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}]$ and $[ \begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix} ]$ of trace equal to $N$, and $Ψ(N)=\sum_{n=3}^N Φ(n)$ be the number of such products of trace between $3$ and $N$. We prove an asymptotic formula of type $Ψ(N) = c_1 N^2 \log N +c_2 N^2 + O_\varepsilon (N^{7/4+\varepsilon})$ as $N\to \infty$. As a result, the Dirichlet series $\sum_{n=1}^\infty Φ(n) n^{-s}$ has a meromorphic extension in the half-plane $\Re (s)>7/4$ with a single, order two pole at $s=2$. Our estimate also improves on an asymptotic result of Faivre concerning the distribution of reduced quadratic irrationals, providing an explicit upper bound for the error term.

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arXiv2020

Projections in Rotation Algebras and Theta Functions

For each $α\in (0,1)$, $A_α$ denotes the universal $C^*$-algebra generated by two unitaries $u$ and $v$, which satisfy the commutation relation $uv=\exp (2πiα)vu$. We consider the order four automorphism $σ$ of $A_α$ defined by $σ(u)=v$, $σ(v)=u^{-1}$ and describe a method for constructing projections in the fixed point algebra $A_α^σ$, using Rieffel's imprimitivity bimodules and Jacobi's theta functions. In the case $α=q^{-1}$, $q\in {\mathbf Z}$, $q\geq 2$, we give explicit formulae for such projections and find a lower bound for the norm of the Harper operator $u+u^* +v+v^*$.

math.OA↗
arXiv2019

Coding of geodesics on some modular surfaces and applications to odd and even continued fractions

The connection between geodesics on the modular surface $\operatorname{PSL}(2,{\mathbb Z})\backslash {\mathbb H}$ and regular continued fractions, established by Series, is extended to a connection between geodesics on $Γ\backslash {\mathbb H}$ and odd and grotesque continued fractions, where $Γ\cong {\Bbb Z}_3 \ast {\Bbb Z}_3$ is the index two subgroup of $\operatorname{PSL}(2,{\mathbb Z})$ generated by the order three elements $\left( \begin{smallmatrix} 0 & -1 \\ 1 & 1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 0 & 1 \\ -1 & 1 \end{smallmatrix} \right)$, having an ideal quadrilateral as fundamental domain. A similar connection between geodesics on $Θ\backslash {\mathbb H}$ and even continued fractions is discussed in our framework, where $Θ$ denotes the Theta subgroup of $\operatorname{PSL}(2,{\mathbb Z})$ generated by $\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix} \right)$.

math.DS↗
arXiv2019

$α$-Expansions with odd partial quotients

We consider an analogue of Nakada's $α$-continued fraction transformation in the setting of continued fractions with odd partial quotients. More precisely, given $α\in [\frac{1}{2}(\sqrt{5}-1),\frac{1}{2}(\sqrt{5}+1)]$, we show that every irrational number $x\in I_α=[α-2,α)$ can be uniquely represented as $$ x= \cfrac{e_1 (x;α)}{d_1 (x;α) +\cfrac{e_2(x;α)}{d_2(x;α)+\cdots}} , $$ with $e_i(x;α) \in \{ \pm 1\}$ and $d_i(x;α) \in 2{\mathbb N} -1$ determined by the iterates of the transformation $$φ_α(x) := \frac{1}{| x|} - 2 \bigg[ \frac{1}{2| x|} +\frac{1-α}{2} \bigg]-1$$ of $I_α$. We also describe the natural extension of $φ_α$ and prove that the endomorphism $φ_α$ is exact.

math.DS↗
arXiv2018

Limiting distribution of eigenvalues in the large sieve matrix

The large sieve inequality is equivalent to the bound $λ_1 \leqslant N + Q^2-1$ for the largest eigenvalue $λ_1$ of the $N$ by $N$ matrix $A^{\star} A$, naturally associated to the positive definite quadratic form arising in the inequality. For arithmetic applications the most interesting range is $N \asymp Q^2$. Based on his numerical data Ramaré conjectured that when $N \sim αQ^2$ as $Q \rightarrow \infty$ for some finite positive constant $α$, the limiting distribution of the eigenvalues of $A^{\star} A$, scaled by $1/N$, exists and is non-degenerate. In this paper we prove this conjecture by establishing the convergence of all moments of the eigenvalues of $A^{\star} A$ as $Q\rightarrow\infty$. Previously only the second moment was known, due to Ramaré. Furthermore, we obtain an explicit description of the moments of the limiting distribution, and establish that they vary continuously with $α$. Some of the main ingredients in our proof include the large-sieve inequality and results on $n$-correlations of Farey fractions.

math.NT↗
arXiv2015

A note on full free product C*-algebras, lifting and quasidiagonality

We study lifting properties for full product C*-algebras with amalgamation over ${\mathbb C}1$ and give new proofs for some results of Kirchberg and Pisier. We extend the result of Choi on the quasidiagonality of $C^*({\mathbb F}_n)$, proving that the free product with amalgamation over ${\mathbb C}1$ of a family of unital quasidiagonal C*-algebras is quasidiagonal.

math.OA↗
arXiv2014

Pair correlation of hyperbolic lattice angles

Let $ω$ be a point in the upper half plane, and let $Γ$ be a discrete, finite covolume subgroup of $\mathrm{PSL}_2(\mathbb{R})$. We conjecture an explicit formula for the pair correlation of the angles between geodesic rays of the lattice $Γω$, intersected with increasingly large balls centered at $ω$. We prove this conjecture for $Γ=\mathrm{PSL}_2(\mathbb{Z})$ and $ω$ an elliptic point.

math.NT↗
arXiv2013

Pair correlation of angles between reciprocal geodesics on the modular surface

The existence of the limiting pair correlation for angles between reciprocal geodesics on the modular surface is established. An explicit formula is provided, which captures geometric information about the length of reciprocal geodesics, as well as arithmetic information about the associated reciprocal classes of binary quadratic forms. One striking feature is the absence of a gap beyond zero in the limiting distribution, contrasting with the analog Euclidean situation.

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