arXiv subjects

Tolibjon Ismoilov

Publications and source records attributed to Tolibjon Ismoilov.

5 recordsLinked to original sources

arXiv2026

Sharp sign uncertainty for trigonometric polynomials

We study sign uncertainty principles for trigonometric polynomials of prescribed degree $N$ with respect to a symmetric Borel measure $\mu$ on the unit circle $\mathbb{R}/\mathbb{Z}$. For each such measure, we determine the smallest radius of the last sign change for trigonometric polynomials with non-positive $\mu$-integral. We further extend these results to polar measures on higher-dimensional spheres $\mathbb{S}^d$, showing that the extremal problem reduces to the one-dimensional case via the polar part of the measure, and we establish a polynomial analogue on $[0,1]$ using orthogonal polynomials on the real line.

math.CA↗

arXiv2025

Littlewood's estimates for $L$-functions in the hyperelliptic ensemble

We investigate the analogues of certain classical estimates of Littlewood for the Riemann zeta-function in the context of quadratic Dirichlet $L$-functions over function fields. In some situations, we are actually able to establish finer results in the function field setup than what is currently known in the original number field setup, and this leads us to an educated guess on what could happen for the Riemann zeta-function in such situations. Fourier analysis techniques play an important role in our approach.

math.NT↗

arXiv2025

Maximizers of the $L^2\to L^4$ Fourier extension inequality for cones in finite fields

Sharp Fourier restriction theory and finite field extension theory have both been topics of interest in the last decades. Very recently, in \cite{GonzalezOliveira}, the research into the intersection of these two topics started. There it was established that, for the $(3,1)$-cone $\Gamma_{(3,1)}^3:=\{\boldsymbol{\eta}\in \mathbb{F}_q^4\setminus\{\boldsymbol{0}\} : \eta_1^2+\eta_2^2+\eta_3^2=\eta_4^2\},$ the Fourier extension map from $L^2\to L^{4}$ is maximized by constant functions when $q=3\, \pmod{4}$. In this manuscript, we advance this line of inquiry by establishing sharp inequalities for the $L^{2}\to L^{4}$ extension inequalities applicable for all remaining cones $\Gamma^3\subset \mathbb{F}_q^4$. These cones include the $(2,2)$-cone $\Gamma_{(2,2)}^3:=\{\boldsymbol{\eta}\in \mathbb{F}_q^4\setminus\{\boldsymbol{0}\} : \eta_1^2+\eta_2^2=\eta_3^2+\eta_4^2\}$ for general $q=p^n$ and the $(3,1)$-cone when $q=1\, \pmod{4}$. Moreover, we classify all the extremizers in each case. We note that the analogous problem for the $(2, 2)$-cone in the euclidean setting remains open.

math.CA↗

arXiv2025

Fourier optimization and pair correlation problems

We introduce a generic framework to provide bounds related to the pair correlation of sequences belonging to a wide class. We consider analogues of Montgomery's form factor for zeros of the Riemann zeta function in the case of arbitrary sequences satisfying some basic assumptions, and connect their estimation to two extremal problems in Fourier analysis, which are promptly studied. As applications, we provide average bounds of form factors related to some sequences of number theoretic interest, such as the zeros of primitive elements of the Selberg class, Dedekind zeta functions, and the real and imaginary parts of the Riemann zeta function. In the last case, our results bear an implication to a conjecture of Gonek and Ki (2018), showing it cannot hold in some situations.

math.NT↗

arXiv2024

Sign uncertainty and de Branges spaces

We investigate here the sign uncertainty phenomenon for bandlimited functions, with a competing condition given by integration with respect to a general measure. Our main result provides a framework related to the theory of de Branges spaces of entire functions, that allows one to find the sharp constants and classify the extremizers in a broad range of situations. We discuss an application in number theory, in connection to bounds for zeros of $L$-functions.

math.CA↗