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Francesco Cellarosi

Publications and source records attributed to Francesco Cellarosi.

At least 19 recordsLinked to original sources
arXiv2026

Density of large holes among power-free lattice points

For fixed integers $d,r\ge1$ with $dr\ge2$, the $r$-free points of $\mathbb{Z}^d$ are the vectors whose coordinate gcd is not divisible by the $r$th power of any prime. We consider the set $W_{d,r}$ of non-$r$-free points in $\mathbb{Z}^d$ as the vertex set of a graph with nearest-neighbour adjacency. For $1\le q\le\infty$, we say that a point $t\in\mathbb Z^d$ is $R$-deep if $t+\{n\in\mathbb Z^d:\|n\|_q\le R\}\subseteq W_{d,r}$. Thus $R$-deep points are the lattice centres of closed $\ell_q$-balls whose lattice points are all non-$r$-free. We call a connected component $R$-large if it contains an $R$-deep point. We mark each finite $R$-large component by selecting its lexicographically least $R$-deep point as its representative. Uniformly for $1\le q\le\infty$, we show that the density of $R$-deep points and the density of these representatives both equal $$ \exp\!\left\{ -\frac{v_{d,q}}{ζ(dr)} \left(d(dr-1)R^d\log R+drR^d\log\log R\right) +O_{d,r}(R^d) \right\} $$ as $R\to\infty$, where $v_{d,q}$ denotes the volume of the unit ball in $(\mathbb R^d,\|\cdot\|_q)$. For deep points, this sharpens the positive-density hole constructions of Baake, Moody, and Pleasants and of Pleasants and Huck, and the latter authors' upper bounds for sparse-pattern frequencies. When $d=1$, we show that the density of $r$-free integers followed by exactly $g-1$ non-$r$-free integers and then another $r$-free integer is $$\exp\!\left\{-\frac{r-1}{ζ(r)}g\log g -\frac{r}{ζ(r)}g\log\log g+O_r(g)\right\}$$ as $g\to\infty$, improving the asymptotic $\exp\left\{-\left(\frac{r-1}{ζ(r)}+o(1)\right) g\log g\right\}$ which follows from Grimmett's work. % For fixed $r$, we also obtain estimates uniform in growing dimensions $d=O_r((\log R)^r)$.

math.NT↗
arXiv2026

Intrinsic ergodicity for $\mathfrak{B}$-free integers in number fields

Let $K$ be a number field with ring of integers $\mathscr{O}_K$, and let $\mathfrak{B}$ be an Erdős family of ideals in $\mathscr{O}_K$. We prove that the associated $\mathfrak{B}$-free subshift $(X_{\mathfrak{B}},(S_a)_{a\in\mathscr{O}_K})$ is intrinsically ergodic: it carries a unique measure of maximal entropy, which we identify explicitly as a relatively independent extension of the Haar rotation on $\prod_{\mathfrak{b}\in\mathfrak{B}}\mathscr{O}_K/\mathfrak{b}$. This is the first proof of intrinsic ergodicity for $\mathfrak{B}$-free systems beyond dimension one, and relies on the work of Araújo--Dymek--Kułaga-Przymus. Via their reductions, we also settle the $k$-free and $\mathfrak{B}$-free lattice-point cases and the $k$-free number-field case. We give two independent proofs of the underlying rigidity statement: one through a single-site relative-entropy argument, and one through an exact-tiling realisation of Peckner's induce-and-split scheme.

math.DS↗
arXiv2026

Continuous dependence results for quasilinear evolution equations

We study continuous dependence of solutions to quasilinear evolution equations of parabolic-type in the framework of maximal $L^p$-regularity. For equations of the form \[ \frac{dϕ}{dt} + A(t,ϕ)ϕ= f(t,ϕ), \] we establish continuous dependence of strong solutions on initial data, and suitable approximations of the nonlinear operators $A$ and $f$. An important step for proving the main result is the fact that the maximal regularity constant of the operator $A(t,ϕ)$, with $t$ and $ϕ$ fixed, admits a uniform bound over compact subsets of the relevant Banach spaces. As an application, we consider a class of non-Newtonian fluid models with a Carreau-type viscosity and mixed boundary conditions. We show that, as the nonlinear contribution in the viscosity vanishes and the initial data converge, solutions of the non-Newtonian fluid model converge to those of the classical Navier--Stokes equations.

math.AP↗
arXiv2025

Asymptotic stability of solutions to semilinear evolution equations in Banach spaces

We prove a new linearization principle for the nonlinear stability of solutions to semilinear evolution equations of parabolic type. We assume that the set of equilibria forms a finite dimensional manifold of normally stable and normally hyperbolic equilibria. In addition, we assume that the linearized operator is the generator of an analytic semigroup (not necessarily stable). We show that if a mild solution to our evolution equation exists globally in time and remains ``close'' to the manifold of equilibria at all times, then the solution must eventually converge to an equilibrium point at an exponential rate. We apply our abstract results to the equations governing the motion of a fluid-filled heavy solid. Under general assumptions on the physical configuration and initial conditions, we show that weak solutions to the governing equations eventually converge to a steady state with an exponential rate. In particular, the fluid velocity relative to the solid converges to zero as $t\to\infty$ in $H^{2α}_p(Ω)$ for each $p\in [1,\infty)$ and $α\in [0,1)$ as well as in $H^{2}_2(Ω)$.

math.AP↗
arXiv2025

Stochastic Calculus for the Theta Process

The theta process is a stochastic process of number theoretical origin arising as a scaling limit of quadratic Weyl sums. It can be described in terms of the geodesic flow and an automorphic function on a homogeneous space. This process has several properties in common with Brownian motion such as its Hölder regularity, uncorrelated increments and quadratic variation. However, crucially, we show that the theta process is not a semimartingale, making Itô calculus techniques inapplicable. Instead, we use the celebrated rough paths theory to develop the stochastic calculus for the theta process. We do so by constructing the iterated integrals - the ``rough path" - above the theta process. Rough paths theory takes a signal and its iterated integrals and produces a vast and robust theory of stochastic differential equations. In addition, the rough path we construct can be described in terms of higher rank theta sums, via equidistribution of horocycle lifts.

math.PR↗
arXiv2024

Rough Paths above Weierstrass Functions

Rough paths theory allows for a pathwise theory of solutions to differential equations driven by highly irregular signals. The fundamental observation of rough paths theory is that if one can define "iterated integrals" above a signal, then one can construct solutions to differential equations driven by the signal. The typical examples of the signals of interest are stochastic processes such as (fractional) Brownian motion. However, rough paths theory is not inherently random and therefore can treat irregular deterministic driving signals such as a (multivariate) Weierstrass function. To the authors' best knowledge, no explicit construction of a rough path (the "iterated integrals") above a multivariate Weierstrass function has been constructed, nor has there been an explicit solution to a differential equation driven by a multivariate Weierstrass function. This note supplies a construction of a rough path above a multivariate Weierstrass function. We conclude with some illustrations and some examples of solving differential equations driven by rough Weierstrass functions.

math.DS↗
arXiv2023

Bounds for Smooth Theta Sums with Rational Parameters

We provide an explicit family of pairs $(α, β) \in \mathbb{R}^k \times \mathbb{R}^k$ such that for sufficiently regular $f$, there is a constant $C>0$ for which the theta sum bound $$\left|\sum_{n\in\mathbb{Z}^k}f\!\left(\tfrac{1}{N}n\right)\exp\left\{2πi\left(\left(\tfrac{1}{2}\|n\|^2+β\cdot n\right)x+α\cdot n\right)\right\}\right|\leq C N^{k/2}$$ holds for every $x \in \mathbb{R}$ and every $N \in \mathbb{N}$. Central to the proof is realising that, for fixed $N$, the theta sum normalised by $N^{k/2}$ agrees with an automorphic function $|Θ_f|$ evaluated along a special curve known as a horocycle lift. The lift depends on the pair $(α,β)$, and so the bound follows from showing that there are pairs such that $|Θ_f|$ remains bounded along the entire horocycle lift.

math.NT↗
arXiv2023

Improved Tail Estimates for the Distribution of Quadratic Weyl Sums

We consider quadratic Weyl sums $S_N(x;c,α)=\sum_{n=1}^N\exp\{2πi((\frac{1}{2}n^2+cn)x+αn)\}$ for $c=α=0$ (the rational case) or $(c,α)\notin\mathbb{Q}^2$ (the irrational case), where $x$ is randomly distributed according to a probability measure absolutely continuous with respect to the Lebesgue measure. The limiting distribution in the complex plane of $\frac{1}{\sqrt{N}}S_N(x;c,α)$ as $N\to\infty$ was described by Marklof [13] (respectively Cellarosi and Marklof [5]) in the rational (resp. irrational) case. According to the limiting distribution, the probability of landing outside a ball of radius $R$ is known to be asymptotic to $\frac{4\log 2}{π^2}R^{-4}(1+o(1))$ in the rational case and to $\frac{6}{π^2}R^{-6}(1+O(R^{-12/31}))$ in the irrational case, as $R\to\infty$. In this work we refine the technique of Cellarosi and Marklof [5] to improve the known tail estimates to $\frac{4\log 2}{π^2}R^{-4}(1+O_\varepsilon(R^{-2+\varepsilon}))$ and $\frac{6}{π^2}R^{-6}(1+O_\varepsilon(R^{-2+\varepsilon}))$ for every $\varepsilon>0$. In the rational case, we rely on the equidistribution of a rational horocycle lift to a torus bundle over the unit tangent bundle to the classical modular surface. All the constants implied by the $O_\varepsilon$-notations are made explicit

math.NT↗
arXiv2023

Heavy tailed and compactly supported distributions of quadratic Weyl sums with rational parameters

We consider quadratic Weyl sums $S_N(x;α,β)=\sum_{n=1}^N \exp\!\left[2πi\left( \left(\tfrac{1}{2}n^2+βn\right)\!x+αn\right)\right]$ for $(α,β)\in\mathbb{Q}^2$, where $x\in\mathbb{R}$ is randomly distributed according to a probability measure absolutely continuous with respect to the Lebesgue measure. We prove that the limiting distribution in the complex plane of $\frac{1}{\sqrt{N}}S_N(x;α,β)$ as $N\to\infty$ is either heavy tailed or compactly supported, depending solely on $α,β$. In the heavy tailed case, the probability (according to the limiting distribution) of landing outside a ball of radius $R$ is shown to be asymptotic to $\mathcal{T}(α,β)R^{-4}$, where the constant $\mathcal{T}(α,β)>0$ is explicit. The result follows from an analogous statement for products of generalized quadratic Weyl sums of the form $S_N^f(x;α,β)=\sum_{n\in\mathbb{Z}} f\left(\frac{n}{N}\right)\exp\!\left[2πi\left( \left(\tfrac{1}{2}n^2+βn\right)\!x+αn\right)\right]$ where $f$ is regular. The precise tails of the limiting distribution of $\frac{1}{N}S_N^{f_1}\bar{S_N^{f_2}}(x;α,β)$ as $N\to\infty$ can be described in terms of a measure -- which depends on $(α,β)$ -- of a super level set of a product of two Jacobi theta functions on a noncompact homogenous space. Such measures are obtained by means of an equidistribution theorem for rational horocycle lifts to a torus bundle over the unit tangent bundle to a cover of the classical modular surface. The cardinality and the geometry of orbits of rational points of the torus under the affine action of the theta group play a crucial role in the computation of $\mathcal{T}(α,β)$. This paper complements and extends the works of Cellarosi and Marklof [6] and Marklof [32], where $(α,β)\notin\mathbb{Q}^2$ and $α=β=0$ are considered.

math.NT↗
arXiv2020

Autocorrelation functions for quantum particles in supersymmetric Pöschl-Teller Potentials

We consider autocorrelation functions for supersymmetric quantum mechanical systems (consisting of a fermion and a boson) confined in trigonometric Pöschl-Teller partner potentials. We study the limit of rescaled autocorrelation functions (at random time) as the localization of the initial state goes to infinity. The limiting distribution can be described using pairs of Jacobi theta functions on a suitably defined homogeneous space, as a corollary of the work of Cellarosi and Marklof. A construction by Contreras-Astorga and Fernández provides large classes of Pöschl-Teller partner potentials to which our analysis applies.

math-ph↗
arXiv2015

Quadratic Weyl Sums, Automorphic Functions, and Invariance Principles

Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation here is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion, non-differentiability), although time increments are not independent and the value distribution at each fixed time is distinctly different from a normal distribution.

math.NT↗
arXiv2014

On two conjectures for M&m sequences

In this paper, the recently introduced M&m sequences and associated mean-median map are studied. These sequences are built by adding new points to a set of real numbers by balancing the mean of the new set with the median of the original. This process, although seemingly simple, gives rise to complicated dynamics. The main result is that two conjectures put forward by Chamberland and Martelli are shown to be true for a subset of possible starting conditions.

math.CO↗
arXiv2014

Continued fraction digit averages an Maclaurin's inequalities

A classical result of Khinchin says that for almost all real numbers $α$, the geometric mean of the first $n$ digits $a_i(α)$ in the continued fraction expansion of $α$ converges to a number $K = 2.6854520\ldots$ (Khinchin's constant) as $n \to \infty$. On the other hand, for almost all $α$, the arithmetic mean of the first $n$ continued fraction digits $a_i(α)$ approaches infinity as $n \to \infty$. There is a sequence of refinements of the AM-GM inequality, Maclaurin's inequalities, relating the $1/k$-th powers of the $k$-th elementary symmetric means of $n$ numbers for $1 \leq k \leq n$. On the left end (when $k=n$) we have the geometric mean, and on the right end ($k=1$) we have the arithmetic mean. We analyze what happens to the means of continued fraction digits of a typical real number in the limit as one moves $f(n)$ steps away from either extreme. We prove sufficient conditions on $f(n)$ to ensure to ensure divergence when one moves $f(n)$ steps away from the arithmetic mean and convergence when one moves $f(n)$ steps away from the geometric mean. For typical $α$ we conjecture the behavior for $f(n)=cn$, $0<c<1$. We also study the limiting behavior of such means for quadratic irrational $α$, providing rigorous results, as well as numerically supported conjectures.

math.NT↗
arXiv2013

Ergodic Properties of $k$-Free Integers in Number Fields

Let $K/\mathbf Q$ be a degree $d$ extension. Inside the ring of integers $\mathcal O_K$ we define the set of $k$-free integers $\mathcal F_k$ and a natural $\mathcal O_K$-action on the space of binary $\mathcal O_K$-indexed sequences, equipped with an $\mathcal O_K$-invariant probability measure associated to $\mathcal F_k$. We prove that this action is ergodic, has pure point spectrum and is isomorphic to a $\mathbf Z^d$-action on a compact abelian group. In particular, it is not weakly mixing and has zero measure-theoretical entropy. This work generalizes the paper by the first author and Sinai arXiv:1112.4691 [math.DS] where $K=\mathbf Q$ and $k=2$.

math.DS↗
arXiv2013

Smooth Sums over Smooth $k$-Free Numbers and Statistical Mechanics

We provide an asymptotic estimate for certain sums over k-free integers with small prime factors. These sums depend upon a complex parameter αand involve a smooth cut-off f. They are a variation of several classical number-theoretical sums. One term in the asymptotics is an integral operator whose kernel is the α-convolution of the Dickman-de Bruijn distribution, and the other term is explicitly estimated. The trade-off between the value of αand the regularity of f is discussed. This work generalizes the results of tow previous papers by the author and Ya.G. Sinai, where k=2 and α=1.

math.NT↗
arXiv2013

Ergodic Properties of Square-Free Numbers

We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This implies that this system is not weakly mixing and has zero measure-theoretical entropy.

math.DS↗
arXiv2010

Non-Standard Limit Theorems in Number Theory

We present a limit theorem describing the behavior of a probabilistic model for square-free numbers. The limiting distribution has a density that comes from the Dickman-De Bruijn function and is constant on the interval $[0,1]$. We also provide estimates concerning the error term in the limit theorem.

math.PR↗