arXiv subjects
Publications and source records attributed to Reynold Fregoli.
We investigate the number of integer solutions to a multiplicative Diophantine approximation problem and show that the associated counting function converges in distribution to a normal law. Our approach relies on the analysis of correlations of measures on homogeneous spaces, together with estimates for Siegel transforms restricted to subspaces.
We show that the Hausdorff dimension of $\boldsymbol w$-weighted $\tau$-exactly approximable vectors in $\mathbb R^d$ coincides with the Hausdorff dimension of $\boldsymbol w$-weighted $\tau$-approximable vectors, generalizing a result of the first named author and De Saxc\'e.
We generalize results of Jones and Olsen on multi-parameter moving ergodic averages to measure-preserving actions of $\mathbb R^d$ for $d\geq 1$. In particular, we give necessary and sufficient conditions for the pointwise convergence of averages over families of boxes in $\mathbb R^d$. As an application of our characterization, we show that averages along dilates of "locally flat" submanifolds in $\mathbb R^d$ do not necessarily converge point-wise for bounded measurable functions. This is closely related to the concept of submanifold-genericity recently introduced in \cite{BFK25}.
In this paper, we prove a new ergodic theorem for $\mathbb{R}^d$-actions involving averages over dilated submanifolds, thereby generalizing the theory of spherical averages. Our main result is a quantitative estimate for the error term of such averages valid for smooth functions under some effective mixing assumptions on the action. With the aid of this theorem, we investigate multiplicative-type Dirichlet-improvability for $(m\times n)$-matrices with real coefficients. In particular, we establish that almost all matrices are uniformly approximable by the function $x\mapsto x^{-1}(\log x)^{-1+\varepsilon}$ for any $\varepsilon>0$. Results of this type motivate a question which can be thought as a strengthening of Littlewood's conjecture in multiplicative Diophantine approximation.
A profound link between Homogeneous Dynamics and Diophantine Approximation is based on an observation that Diophantine properties of a real matrix $B$ are encoded by the corresponding lattice $\Lambda_B$ translated by a multi-parameter semigroup $a(t)$. We establish quantitative decorrelation estimates for measures supported on leaves $a(t)\Lambda_B$ with the error terms depending only on the minimum of the pairwise distances between the parameters. The proof involves a careful analysis of the translated measures in the products of the spaces of unimodular lattices and establishes quantitative equidistributions to measures supported on various intermediate homogeneous subspaces.
We prove an estimate for the number of lattice points lying in certain non-convex Euclidean domains of interest in Diophantine approximation. As an application, we generalise a result of Kruse (1964) concerning the almost sure order of magnitude of sums of reciprocals of fractional parts and solve a conjecture posed by Beresnevich, Haynes, and Velani. The methods are based both on the geometry of numbers and on probability theory.
We compute the Hausdorff dimension of the set of $\psi$-exactly approximable vectors, in the simultaneous case, in dimension strictly larger than $2$ and for approximating functions $\psi$ with order at infinity less than or equal to $-2$. Our method relies on the analogous result in dimension $1$, proved by Yann Bugeaud and Carlos Moreira, and a version of Jarn\'ik's Theorem on fibres.
Let $\langle x\rangle$ denote the distance from $x\in\mathbb{R}$ to the set of integers $\mathbb{Z}$. The Littlewood Conjecture states that for all pairs $(\alpha,\beta)\in\mathbb{R}^{2}$ the product $q\langle q\alpha\rangle\langle q\beta\rangle$ attains values arbitrarily close to $0$ as $q\in\mathbb{N}$ tends to infinity. Badziahin showed that if a factor $\log q\cdot \log\log q$ is added to the product, the same statement becomes false. In this paper, we generalise Badziahin's result to vectors $\boldsymbol{\alpha}\in\mathbb{R}^{d}$, replacing the function $\log q\cdot \log\log q$ by $(\log q)^{d-1}\cdot\log\log q$ for any $d\geq 2$, and thereby obtaining a new proof in the case $d=2$. Our approach is based on a new version of the well-known Dani Correspondence between Diophantine approximation and dynamics on the space of lattices, especially adapted to the study of products of rational approximations. We believe that this correspondence is of independent interest.
In this paper, we study the shrinking-target problem with target at infinity induced by the injectivity radius function under the action of a regular diagonalizable flow on $\operatorname{SL}_3(\mathbb R)/\operatorname{SL}_3(\mathbb Z)$. In particular, we establish an explicit formula for the Hausdorff dimension of the subset of points $p$ whose orbit approaches the cusp infinitely often with a rate $\gamma\geq0$.
For Lebesgue generic $(x_1,x_2)\in \mathbb{R}^2$, we investigate the distribution of small values of products $q\cdot \|qx_1\| \cdot \|qx_2\|$ with $q\in\mathbb{N}$, where $\|\cdot \|$ denotes the distance to the closest integer. The main result gives an asymptotic formula for the number of $1\le q\le T$ such that $$ a_T <q\cdot \|qx_1\| \cdot \|qx_2\|\leq b_T \quad \textrm{and} \quad \|qx_1\|, \|qx_2\|\leq c_T $$ for given sequences $a_T,b_T, c_T$ satisfying certain growth conditions.
In this paper, we prove new upper bounds for sums of reciprocals of fractional parts over general aligned boxes, thus extending a previous result of the author concerning bounds for sums of reciprocals over symmetric boxes. These new upper bounds depend solely on the volume of the boxes, and not on their diameter. This generalisation relies a novel lattice-point counting technique involving estimates for the higher successive minima of certain naturally arising lattices.
Let $\|x\|$ denote the distance from $x\in\mathbb{R}$ to the nearest integer. In this paper, we prove an existence and density statement for matrices $\boldsymbol{A}\in\mathbb{R}^{m\times n}$ satisfying $$\liminf_{|\boldsymbol{q}|_{\infty}\to +\infty}\prod_{j=1}^{n}\max\{1,|q_{j}|\}\log\left(\prod_{j=1}^{n}\max\{1,|q_{j}|\}\right)^{m+n-1}\prod_{i=1}^{m}\|A_{i}\boldsymbol{q}\|>0,$$ where the vector $\boldsymbol{q}$ ranges in $\mathbb{Z}^{n}$ and $A_{i}$ are the rows of the matrix $\boldsymbol{A}$. This result extends a previous result of Moshchevitin for $2$-dimensional vectors to arbitrary dimension. The estimates needed to apply Moshchevitin's method to the case $m>2$ are not currently available. We therefore develop a substantially different method, that allows us to overcome this issue. We also generalise this existence result to the inhomogeneous setting. Matrices with the above property appear to have a very small sum of reciprocals of fractional parts. This fact helps us to shed light on a question raised by Lê and Vaaler, thereby proving some new estimates for such sums in higher dimension.
Let $A\subset\mathbb{N}$, $α\in(0,1)$, and for $x\in\mathbb{R}$ let $e(x):=e^{2πix}$. We set $$S_{A}(α,N):=\sum_{\substack{n\in A\n\leq N}}e(nα).$$ Recently, Lambert A'Campo proposed the following question: is there an infinite non-cofinite set $A\subset\mathbb{N}$ such that for all $α\in(0,1)$ the sum $S_{A}(α,N)$ has bounded modulus as $N\to +\infty$? In this note we show that such sets do not exist. To do so, we use a theorem by Duffin and Schaeffer on complex power series. We extend our result by proving that if the sum $S_{A}(α,N)$ is bounded in modulus on an arbitrarily small interval and on the set of rational points, then the set $A$ has to be either finite or cofinite. On the other hand, we show that there are infinite non-cofinite sets $A$ such that $|S_{A}(α,N)|$ is bounded for all $α\in E\subset (0,1)$, where $E$ has full Hausdorff dimension and $\mathbb{Q}\cap (0,1)\subset E$.
We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve `hyperbolic spikes' and occur naturally in multiplicative Diophantine approximation. We use Wilkie's o-minimal structure $\mathbb{R}_{\exp}$ and expansions thereof to formulate our counting result in a general setting. We give two different applications of our counting result. The first one establishes nearly sharp upper bounds for sums of reciprocals of fractional parts, and thereby sheds light on a question raised by Lê and Vaaler, extending previous work of Widmer and of the author. The second application establishes new examples of linear subspaces of Khintchine type thereby refining a theorem by Huang and Liu. For the proof of our counting result we develop a sophisticated partition method which is crucial for further upcoming work on sums of reciprocals of fractional parts over distorted boxes.
Let $\boldsymbolα\in \mathbb{R}^N$ and $Q\geq 1$. We consider the sum $\sum_{\boldsymbol{q}\in [-Q,Q]^N\cap\mathbb{Z}^N\backslash\{\boldsymbol{0}\}}\|\boldsymbolα\cdot\boldsymbol{q}\|^{-1}$. Sharp upper bounds are known when $N=1$, using continued fractions or the three distance theorem. However, these techniques do not seem to apply in higher dimension. We introduce a different approach, based on a general counting result of Widmer for weakly admissible lattices, to establish sharp upper bounds for arbitrary $N$. Our result also sheds light on a question raised by Lê and Vaaler in 2013 on the sharpness of their lower bound $\gg Q^N\log Q$.