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Mikhail Sodin

Publications and source records attributed to Mikhail Sodin.

At least 19 recordsLinked to original sources
arXiv2026

A measurable equivariant Weierstrass theorem

This paper is a prequel to our recent work, "Equivariant Borel liftings in complex analysis and PDE" (arXiv:2507.12058). While the results presented here were established in that work in a more general and abstract setting, the purpose of this paper is to provide a direct proof of the equivariant Weierstrass theorem. It states that there exists a Borel map assigning to each non-periodic positive divisor $\Lambda$ an entire function $F_\Lambda$ such that the divisor of zeroes of $F_\Lambda$ is $\Lambda$ and such that $F_{\Lambda-w}(z) = F_\Lambda (z+w)$, $w\in\mathbb{C}$. In general, non-periodicity cannot be omitted, and Borel measurability cannot be strengthened to continuity. The two key ingredients are the Runge approximation theorem and the existence of "Borel toasts", which are Borel counterparts of Rokhlin towers from ergodic theory. We do not assume prior knowledge of descriptive set theory and have aimed to make the exposition self-contained, aside from several results taken from graduate textbooks.

math.CV↗
arXiv2026

When a meromorphic function that omits three values is of bounded type

Suppose that a function $F$ is meromorphic in the domain $\mathbb H(-m) = \{ z : \mathrm{Im}\, z > -m(\mathrm{Re}\, z) \}$, where $m$ is an even, positive, and continuous function that does not increase on $\mathbb R_{\ge 0}$, and suppose that $F$ omits there three distinct values. Then $F$ is of bounded type in the upper half-plane (i.e., is represented there as a quotient of two bounded analytic functions), provided that the logarithmic integral of the function $m$ is convergent. On the other hand, if the logarithmic integral of $m$ diverges, there exists a function $F$ meromorphic in $\mathbb H(-m)$, that omits there three distinct values, and which is of unbounded type in the upper half-plane. This result is motivated by a century old question originating with Rolf Nevanlinna.

math.CV↗
arXiv2025

Equivariant Borel liftings in complex analysis and PDE

We establish Borel equivariant analogues of several classical theorems from complex analysis and PDE. The starting point is an equivariant Weierstrass theorem for entire functions: there exists a Borel mapping which assigns to each non-periodic positive divisor $d$ an entire function $f_d$ with divisor of zeros $\mathrm{div}(f_d)=d$ and which commutes with translation, $f_{d-w}(z)=f_d(z+w)$. We also examine the existence of equivariant Borel right inverses for the distributional Laplacian, the heat operator, and the $\bar{\partial}$-operator on the space of smooth functions. We demonstrate that Borel equivariant inverses for these maps exist on the free part of the range. In general, the freeness assumptions cannot be omitted and Borelness cannot be strengthened to continuity. Our positive results follow from a theorem establishing sufficient conditions for the existence of equivariant Borel liftings. Two key ingredients are Runge-type approximation theorems and the existence of Borel toasts, which are Borel analogues of Rokhlin towers from ergodic theory.

math.DS↗
arXiv2025

Taylor coefficients and zeroes of entire functions of exponential type

Let $F$ be an entire function of exponential type represented by the Taylor series \[ F(z) = \sum_{n\ge 0} \omega_n \frac{z^n}{n!} \] with unimodular coefficients $|\omega_n|=1$. We show that either the counting function $n_F(r)$ of zeroes of $F$ grows linearly at infinity, or $F$ is an exponential function. The same conclusion holds if only a positive asymptotic proportion of the coefficients $\omega_n$ is unimodular. This significantly extends a classical result of Carlson (1915). The second result requires less from the coefficient sequence $\omega$, but more from the counting function of zeroes $n_F$. Assuming that $0<c\le |\omega_n| \le C <\infty$, $n\in\mathbb Z_+$, we show that $n_F(r) = o(\sqrt{r})$ as $r\to\infty$, implies that $F$ is an exponential function. The same conclusion holds if, for some $\alpha<1/2$, $n_F(r_j)=O(r_j^{\alpha})$ only along a sequence $r_j\to\infty$. Furthermore, this conclusion ceases to hold if $n_F(r)=O(\sqrt r)$ as $r\to\infty$.

math.CV↗
arXiv2024

Iossif Ostrovskii's work on entire functions

The theory of entire functions and its applications were at the center of Ostrovskii's research interests throughout his entire career. He made lasting contributions to several aspects of this theory, and many of his works had a significant influence on subsequent research. In this note, we describe some of this work.

math.CV↗
arXiv2023

The random Weierstrass zeta function I. Existence, uniqueness, fluctuations

We describe a construction of random meromorphic functions with prescribed simple poles with unit residues at a given stationary point process. We characterize those stationary processes with finite second moment for which, after subtracting the mean, the random function becomes stationary. These random meromorphic functions can be viewed as random analogues of the Weierstrass zeta function from the theory of elliptic functions, or equivalently as electric fields generated by an infinite random distribution of point charges.

math.PR↗
arXiv2023

The random Weierstrass zeta function II. Fluctuations of the electric flux through rectifiable curves

Consider a random planar point process whose law is invariant under planar isometries. We think of the process as a random distribution of point charges and consider the electric field generated by the charge distribution. In Part I of this work, we found a condition on the spectral side which characterizes when the field itself is invariant with a well-defined second-order structure. Here, we fix a process with an invariant field, and study the fluctuations of the flux through large arcs and curves in the plane. Under suitable conditions on the process and on the curve, denoted $Γ$, we show that the asymptotic variance of the flux through $R\,Γ$ grows like $R$ times the signed length of $Γ$. As a corollary, we find that the charge fluctuations in a dilated Jordan domain is asymptotic with the perimeter, provided only that the boundary is rectifiable. The proof is based on the asymptotic analysis of a closely related quantity (the complex electric action of the field along a curve). A decisive role in the analysis is played by a signed version of the classical Ahlfors regularity condition.

math.PR↗
arXiv2023

Fourier uniqueness and non-uniqueness pairs

Motivated by recent works by Radchenko and Viazovska and by Ramos and Sousa, we find sufficient conditions for a pair of discrete subsets of the real line to be a uniqueness or a non-uniqueness pair for the Fourier transform. These conditions are close to each other. The uniqueness result can be upgraded to an interpolation formula, which in turn produces an abundance of discrete measures with discrete Fourier transform.

math.CA↗
arXiv2021

Zero distribution of power series and binary correlation of coefficients

We study the distribution of zeroes of power series with infinite radius of convergence. The coefficients of the series have the form $\xi(n)a(n)$, where $a$ is a smooth sequence of positive numbers, and $\xi$ is a sequence of complex-valued multipliers having binary correlations and no gaps in the spectrum. We show that under certain assumptions on the smoothness of the sequence $a$ and on the binary correlations of the multipliers $\xi$, the zeroes of the power series are equidistributed with respect to a radial measure defined by the sequence $a$. We apply our approach to several examples of the sequence $\xi$: (i) IID sequences, (ii) sequences $e(\alpha n^2)$ with Diophantine $\alpha$, (iii) random multiplicative sequences, (iv) the Golay--Rudin--Shapiro sequence, (v) the indicator function of the square-free integers, (vi) the Thue--Morse sequence.

math.CV↗
arXiv2020

Fluctuations in the number of nodal domains

We show that the variance of the number of connected components of the zero set of the two-dimensional Gaussian ensemble of random spherical harmonics of degree n grows as a positive power of n. The proof uses no special properties of spherical harmonics and works for any sufficiently regular ensemble of Gaussian random functions on the two-dimensional sphere with distribution invariant with respect to isometries of the sphere. Our argument connects the fluctuations in the number of nodal lines with those in a random loop ensemble on planar graphs of degree four, which can be viewed as a step towards justification of the Bogomolny-Schmit heuristics.

math.PR↗
arXiv2019

Notes on the Szego minimum problem. II. Singular measures

In this part, we prove several quantitative results concerning with the Szego minimum problem for classes of measure on the unit circle concentrated on small subsets. As a by-product, we refute one conjecture of Nevai. This note can be read independently from the first one.

math.CV↗
arXiv2019

Notes on the Szego minimum problem. I. Measures with deep zeroes

The classical Szego polynomial approximation theorem states that the polynomials are dense in the space $L^2(ρ)$, where $ρ$ is a measure on the unit circle, if and only if the logarithmic integral of the measure $ρ$ diverges. In this note we give a quantitative version of Szego's theorem in the special case when the divergence of the logarithmic integral is caused by deep zeroes of the measure $ρ$ on a sufficiently rare subset of the circle.

math.CV↗
arXiv2019

The "pits effect" for entire functions of exponential type and the Wiener spectrum

Given a sequence $\xi\colon \mathbb Z_+ \to \mathbb C$, we find a simple spectral condition which guarantees the angular equidistribution of the zeroes of the Taylor series \[ F_\xi (z) = \sum_{n\ge 0} \xi (n) \frac{z^n}{n!}\,. \] This condition yields practically all known instances of random and pseudo-random sequences $\xi$ with this property (due to Nassif, Littlewood, Chen-Littlewood, Levin, Eremenko-Ostrovskii, Kabluchko-Zaporozhets, Borichev-Nishry-Sodin), and provides several new ones. Among them are Besicovitch almost periodic sequences and multiplicative random sequences. It also conditionally yields that the M\"obius function $\mu$ has this property assuming "the binary Chowla conjecture".

math.PR↗
arXiv2019

Eigenfunctions with infinitely many isolated critical points

We construct a Riemannian metric on the $ 2 $-dimensional torus, such that for infinitely many eigenvalues of the Laplace-Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points. A minor modification of our construction implies that each of these eigenfunctions has a level set with infinitely many connected components (i.e., a linear combination of two eigenfunctions may have infinitely many nodal domains).

math.SP↗
arXiv2017

A discrete harmonic function bounded on a large portion of $\mathbb{Z}^2$ is constant

An improvement of the Liouville theorem for discrete harmonic functions on $\mathbb{Z}^2$ is obtained. More precisely, we prove that there exists a positive constant $\varepsilon$ such that if $u$ is discrete harmonic on $\mathbb{Z}^2$ and for each sufficiently large square $Q$ centered at the origin $|u|\le 1$ on a $(1-\varepsilon)$ portion of $Q$ then $u$ is constant.

math.CA↗
arXiv2017

Translation-invariant probability measures on entire functions

We study non-trivial translation-invariant probability measures on the space of entire functions of one complex variable. The existence (and even an abundance) of such measures was proven by Benjamin Weiss. Answering Weiss question, we find a relatively sharp lower bound for the growth of entire functions in the support of such measures. The proof of this result consists of two independent parts: the proof of the lower bound and the construction, which yields its sharpness. Each of these parts combines various tools (both classical and new) from the theory of entire and subharmonic functions and from the ergodic theory. We also prove several companion results, which concern the decay of the tails of non-trivial translation-invariant probability measures on the space of entire functions and the growth of locally uniformly recurrent entire and meromorphic functions.

math.CV↗
arXiv2017

Hole probability for zeroes of Gaussian Taylor series with finite radii of convergence

We study a family of random Taylor series $$F(z) = \sum_{n\ge 0} ζ_n a_n z^n$$ with radius of convergence almost surely $1$ and independent identically distributed complex Gaussian coefficients $(ζ_n)$; these Taylor series are distinguished by the invariance of their zero sets with respect to isometries of the unit disk. We find reasonably tight upper and lower bounds on the probability that $F$ does not vanish in the disk $\{|z|\le r\}$ as $r\uparrow 1$. Our bounds take different forms according to whether the non-random coefficients $(a_n)$ grow, decay or remain of the same order. The results apply more generally to a class of Gaussian Taylor series whose coefficients $(a_n)$ display power-law behavior.

math.CV↗