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William Verreault

Publications and source records attributed to William Verreault.

At least 19 recordsLinked to original sources
arXiv2026

Fourier decay of Gaussian multiplicative chaos and boundary geometry

In this paper, we develop a new approach to studying the Fourier transform of Gaussian multiplicative chaos. We determine the Fourier dimension of these random measures on tori for every smooth log-correlated covariance and in every dimension, and prove sharp bounds and exact formulas for chaos on natural classes of bounded sets and hypersurfaces in Euclidean space. These results highlight how boundary geometry and multifractal concentration jointly govern Fourier decay, including regimes in which the boundary strictly reduces the Fourier dimension. In the circle setting, this gives a substantially simpler new proof of a conjecture of Garban and Vargas.

math.PR↗
arXiv2026

Almost sure upper bound for sums of random multiplicative functions and critical chaos

Let $f$ be a Steinhaus or Rademacher random multiplicative function. We use methods from the theory of critical chaos to improve on the best known upper bound for partial sums of random multiplicative functions. In particular, our results imply that for any $\varepsilon>0$, almost surely $$ \Big|\sum_{n\le x}f(n)\Big| \ll_{f,\varepsilon}\sqrt{x}(\log_2x)^{1/4}(\log_3x)^{1+\varepsilon}. $$ This proves in a strong form a conjecture of Harper on large fluctuations of partial sums of random multiplicative functions, and determines the exact corresponding logarithmic exponent.

math.NT↗
arXiv2026

A proof of conjectures of Esterle and Ransford on negative powers of contractions

Building on work of Ransford, we prove that whenever $E$ is a closed subset of the unit circle of Lebesgue measure zero, there exists a positive sequence $u_n\to\infty$ such that if $T$ is a contraction on a Hilbert space with $\sigma(T)\subset E$ and $\|T^{-n}\|=O(u_n)$, then $T$ is unitary. This confirms conjectures of Esterle and Ransford. Our main new idea is a spikes-in-collars principle for positive subharmonic functions.

math.FA↗
arXiv2025

Central limit theorems for random multiplicative functions over function fields

We provide a sufficient characterization for subsets $\mathcal{A}$ of the polynomial ring $\mathbb{F}_q[t]$ for which partial sums of Steinhaus random multiplicative functions approach a complex standard normal distribution. This extends recent work of Soundararajan and Xu to the function field setting. We apply this characterization to deduce central limit theorems in four cases: polynomials in short intervals, polynomials with few prime factors, shifted primes, and rough polynomials. In doing so, we also establish an explicit Hildebrand inequality for smooth polynomials in short intervals, a function field form of Shiu's theorem for multiplicative functions, and an explicit Chebyshev bound for rough polynomials in short intervals.

math.NT↗
arXiv2025

On the minimal length of addition chains

We denote by $\ell(n)$ the minimal length of an addition chain leading to $n$ and we define the counting function $$ F(m,r):=\#\left\{n\in[2^m, 2^{m+1}):\ell(n)\le m+r\right\}, $$ where $m$ is a positive integer and $r\ge 0$ is a real number. We show that for $0< c<\log 2$ and for any $\varepsilon>0$, we have as $m\to \infty$, $$ F\left(m,\frac{cm}{\log m}\right)<\exp\left(cm+\frac{\varepsilon m\log\log m}{\log m}\right) $$ and $$ F\left(m,\frac{cm}{\log m}\right)>\exp\left(cm-\frac{(1+\varepsilon)cm\log\log m}{\log m}\right). $$ This extends a result of Erd\H{o}s which says that for almost all $n$, as $n\to\infty$, $$ \ell(n)=\frac{\log n}{\log 2}+\left(1+o(1)\right)\frac{\log n}{\log \log n}. $$

math.NT↗
arXiv2024

The Ces\`{a}ro operator on local Dirichlet spaces

The family of Ces\`{a}ro operators $\sigma_n^\alpha$, $n \geq 0$ and $\alpha \in [0,1]$, consists of finite rank operators on Banach spaces of analytic functions on the open unit disc. In this work, we investigate these operators as they act on the local Dirichlet spaces $\mathcal{D}_\zeta$. It is well-established that they provide a linear approximation scheme when $\alpha > \frac{1}{2}$, with the threshold value $\alpha = \frac{1}{2}$ being optimal. We strengthen this result by deriving precise asymptotic values for the norm of these operators when $\alpha \leq \frac{1}{2}$, corresponding to the breakdown of approximation schemes. Additionally, we establish upper and lower estimates for the norm when $\alpha > \frac{1}{2}$.

math.FA↗
arXiv2024

Moments of random multiplicative functions over function fields

Granville-Soundararajan, Harper-Nikeghbali-Radziwill, and Heap-Lindqvist independently established an asymptotic for the even natural moments of partial sums of random multiplicative functions defined over integers. Building on these works, we study the even natural moments of partial sums of Steinhaus random multiplicative functions defined over function fields. Using a combination of analytic arguments and combinatorial arguments, we obtain asymptotic expressions for all the even natural moments in the large field limit and large degree limit, as well as an exact expression for the fourth moment.

math.NT↗
arXiv2024

Arithmetic functions at factorial arguments

For various arithmetic functions $f:\mathbb{N} \to \mathbb{R}$, the behavior of $f(n!)$ and that of $\sum_{n\le N} f(n!)$ can be intriguing. For instance, for some functions $f$, we have ${f(n!)=\sum_{k\le n}f(k)}$, for others, we have ${f(n!)=\sum_{p\le n}f(p)}$ (where the sum runs over all the primes $p\le n$). Also, for some $f$, their minimum order coincides with $\lim_{n\to \infty}f(n!)$, for others, it is their maximum order that does so. Here, we elucidate such phenomena and more generally, we embark on a study of $f(n!)$ and of $\sum_{n\le N}f(n!)$ for a wide variety of arithmetical functions $f$. In particular, letting $d(n)$ and $σ(n)$ stand respectively for the number of positive divisors of $n$ and the sum of the positive divisors of $n$, we obtain new accurate asymptotic expansions for $d(n!)$ and $σ(n!)$. Furthermore, setting $ρ_1(n):=\max\{d\mid n:d\le \sqrt n\}$ and observing that no one has yet obtained an asymptotic value for $\sum_{n\le N} ρ_1(n)$ as $N\to \infty$, we show how one can obtain the asymptotic value of $\sum_{n\le N} ρ_1(n!)$.

math.NT↗
arXiv2023

Nonlinear expansions in reproducing kernel Hilbert spaces

We introduce an expansion scheme in reproducing kernel Hilbert spaces, which as a special case covers the celebrated Blaschke unwinding series expansion for analytic functions. The expansion scheme is further generalized to cover Hardy spaces $H^p$, $1<p<\infty$, viewed as Banach spaces of analytic functions with bounded evaluation functionals. In this setting a dichotomy is more transparent: depending on the multipliers used, the expansion of $f \in H^p$ converges either to $f$ in $H^p$-norm or to its projection onto a model space generated by the corresponding multipliers. Some explicit instances of the general expansion scheme, which are not covered by the previously known methods, are also discussed.

math.FA↗
arXiv2023

On the tower factorization of integers

Under the fundamental theorem of arithmetic, any integer $n>1$ can be uniquely written as a product of prime powers $p^a$; factoring each exponent $a$ as a product of prime powers $q^b$, and so on, one will obtain what is called the tower factorization of $n$. Here, given an integer $n>1$, we study its height $h(n)$, that is, the number of "floors" in its tower factorization. In particular, given a fixed integer $k\geq 1$, we provide a formula for the density of the set of integers $n$ with $h(n)=k$. This allows us to estimate the number of floors that a positive integer will have on average. We also show that there exist arbitrarily long sequences of consecutive integers with arbitrarily large heights.

math.NT↗
arXiv2022

Classes of Holomorphic Multicomplex-Valued Functions Generated by Elliptic-Admissible Involutions

We classify and count the real-algebra involutions of the multicomplex algebra $\mathbb{M}\mathbb{C} (n)$ that map each element of its canonical monomial basis to a signed monomial, using a matrix model over $\mathbb F_2$. An \emph{elliptic-admissible pair} consists of such an involution $\sigma$ and a monomial unit $\mathbf{i}\in\mathbb{I}(n)$ satisfying $\mathbf{i^2}=-1$ and $\sigma(\mathbf{i})=-\mathbf{i}$. The fixed algebra of $\sigma$ is a real form for the complex structure $\mathcal{L}_{\mathbf{i}}$ defined by multiplication by $\mathbf{i}$, and the pair yields a Cauchy--Riemann system. We prove that its solution class depends only on $\mathbf{i}$, not on $\sigma$, and is exactly the class of mappings holomorphic with respect to $\mathcal{L}_{\mathbf{i}}$. Multicomplex holomorphy is recovered as the intersection of the classes associated with the elementary generators. We obtain the analogous characterization for anti-holomorphic classes, describe twisted systems intertwining two such complex structures, and prove that every $\mathbf{i}{}$-holomorphic or $\mathbf{i}$-anti-holomorphic mapping is componentwise harmonic (solutions to Laplace's equation).

math.RA↗
arXiv2022

Failure of $L^p$ Symmetry of Zonal Spherical Harmonics

In this paper, we show that the 2-sphere does not exhibit symmetry of $L^p$ norms of eigenfunctions of the Laplacian for $p\geq 6$. In other words, there exists a sequence of spherical eigenfunctions $ψ_n$, with eigenvalues $λ_n\to\infty$ as $n\to\infty$, such that the ratio of the $L^p$ norms of the positive and negative parts of the eigenfunctions does not tend to $1$ as $n\to\infty$ when $p\geq 6$. Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.

math.CA↗
arXiv2022

A counterexample to symmetry of $L^p$ norms of eigenfunctions

We answer a question of Jakobson and Nadirashvili on the asymptotic behavior of the $L^p$ norms of positive and negative parts of eigenfunctions of the Laplacian. More precisely, we show that there exists a sequence of eigenfunctions $ψ_n$ on the flat $d$-torus for $d\geq 3$, with eigenvalues $λ_n\to\infty$ as $n\to\infty$, such that the ratio $\|ψ_nχ_{\{ψ_n>0\}}\|_p / \|ψ_nχ_{\{ψ_n<0\}}\|_p $ does not tend to $1$ as $n\to\infty$ for $1<p\leq \infty$. Our argument is elementary and computer-assisted.

math.SP↗
arXiv2022

Plank theorems and their applications: a survey

Plank problems concern the covering of convex bodies by planks in Euclidean space and are related to famous open problems in convex geometry. In this survey, we introduce plank problems and present surprising applications of plank theorems in various areas of mathematics.

math.MG↗
arXiv2022

MacMahon Partition Analysis: A discrete approach to broken stick problems

We propose a discrete approach to solve problems on forming polygons from broken sticks, which is akin to counting polygons with sides of integer length subject to certain Diophantine inequalities. Namely, we use MacMahon's Partition Analysis to obtain a generating function for the size of the set of segments of a broken stick subject to these inequalities. In particular, we use this approach to show that for $n\geq k\geq 3$, the probability that a $k$-gon cannot be formed from a stick broken into $n$ parts is given by $n!$ over a product of linear combinations of partial sums of generalized Fibonacci numbers, a problem which proved to be very hard to generalize in the past.

math.CO↗
arXiv2021

A conjectural asymptotic formula for multiplicative chaos in number theory

We investigate a special sequence of random variables $A(N)$ defined by an exponential power series with independent standard complex Gaussians $(X(k))_{k \geq 1}$. Introduced by Hughes, Keating, and O'Connell in the study of random matrix theory, this sequence relates to Gaussian multiplicative chaos (in particular "holomorphic multiplicative chaos'' per Najnudel, Paquette, and Simm) and random multiplicative functions. Soundararajan and Zaman recently determined the order of $\mathbb{E}[|A(N)|]$. By constructing an algorithm to calculate $A(N)$ in $O(N^2 \log N)$ steps, we produce computational evidence that their result can likely be strengthened to an asymptotic result with a numerical estimate for the asymptotic constant. We also obtain similar conclusions when $A(N)$ is defined using standard real Gaussians or uniform $\pm 1$ random variables. However, our evidence suggests that the asymptotic constants do not possess a natural product structure.

math.NT↗
arXiv2021

On the probability of forming polygons from a broken stick

Break a stick at random at $n-1$ points to obtain $n$ pieces. We give an explicit formula for the probability that every choice of $k$ segments from this broken stick can form a $k$-gon, generalizing similar work. The method we use can be applied to other geometric probability problems involving broken sticks, which are part of a long-standing class of recreational probability problems with several applications to real-world models.

math.PR↗