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Kristina Oganesyan

Publications and source records attributed to Kristina Oganesyan.

10 recordsLinked to original sources
arXiv2026

Quantitative possibilities for Baxter's theorem in nonlinear Fourier analysis

Baxter's classical theorem for orthogonal polynomials on the unit circle establishes that the linear Fourier coefficients of a measure are in $\ell^1$ if and only if the nonlinear coefficients, i.e., the so-called Verblunsky coefficients, are in $\ell^1$. However, this equivalence is purely qualitative. We explore possible formulations of such quantitative theorems with norm and Lipschitz estimates for the $SU(2)$-valued nonlinear Fourier transform, for which the analog of Baxter's theorem has been recently obtained. In particular, the highlight of this paper is a number-theoretic construction for the NLFT which proves some negative results in this direction. We also prove a positive result and discuss the limitations of Baxter's method.

math.CA↗
arXiv2026

Upper and lower estimates for integer complexity

Let $\|n\|$ stand for the integer complexity of the number $n$, i.e. for the least number of $1$'s needed to write $n$ using arbitrary many additions, multiplications, and parentheses. The two-sided inequality $3\log_3 n\leq\|n\|\leq 3\log_2 n$ for all $n$ is well known and reveals the logarithmic behaviour of the complexity function $\|n\|$. While the lower bound $3\log_3 n$ is attained infinitely many times at powers of $3$, the best upper estimate is still unknown, although there are some improvements of the trivial bound $3\log_2 n$. Besides, for $``$typical$"$ numbers, i.e. for almost all numbers $n$, the better inequality $\|n\|\leq C_{avg}\log n$ holds, where, importantly, $C_{avg}\approx 3.236<\sup_{n} \frac{\|n\|}{\log n}$. We show that in fact $\|n\|\leq C_{avg}\log n+o(\log n)$ as $n\to\infty$, which, in particular, yields that $\limsup\limits_{n\to\infty}\frac{\|n\|}{\log n}\leq C_{avg}$. We also obtain the first nontrivial lower bound $\|n\|\geq 3.06\log_3 n$ for almost all numbers $n$.

math.NT↗
arXiv2024

Quadratic spectral concentration of characteristic functions

It is known that the inequality \begin{align*}\int_{-W/2}^{W/2}|\widehat{f}(ξ)|^2dξ\leq \int_{-W/2}^{W/2}|\widehat{|f|^*}(ξ)|^2dξ\end{align*} between the quadratic spectral concentration of a function and that of its decreasing rearrangement holds for any function $f\in L^2,\;|\text{supp} f|=T,$ if and only if the product $WT$ does not exceed the critical value $\approx 0.884$. We show that by restricting ourselves to characteristic functions we can enlarge this range up to $WT\leq 4/3$. Besides, we establish various properties of minimizers of the difference $\int_{-W/2}^{W/2}|\widehat{χ_A^*}(ξ)|^2dξ-\int_{-W/2}^{W/2}|\widehat{χ_A}(ξ)|^2dξ$ over sets $A$ of finite measure and prove that this difference is non-negative for all $W,T>0$ if $A$ is the union of two intervals. As a corollary, we obtain a sharp (up to a constant) estimate for the $L_2$-norms of non-harmonic trigonometric polynomials with alternating coefficients $\pm 1$.

math.CA↗
arXiv2024

Bounds for the number of multidimensional partitions

We obtain estimates for the number $p_d(n)$ of $(d-1)$-dimensional integer partitions of a number $n$. It is known that the two-sided inequality $C_1(d)n^{1-1/d}<\log p_d(n)< C_2(d)n^{1-1/d}$ is always true and that $C_1(d)>1$ whenever $\log n> 3d$. However, establishing the $``$right$"$ dependence of $C_2$ on $d$ remained an open problem. We show that if $d$ is sufficiently small with respect to $n$, then $C_2$ does not depend on $d$, which means that $\log p_d(n)$ is up to an absolute constant equal to $n^{1-1/d}$. Besides, we provide estimates of $p_d(n)$ for different ranges of $d$ in terms of $n$, which give the asymptotics of $\log p_d(n)$ in each case.

math.CO↗
arXiv2023

Univalence of $T$-symmetric Suffridge type polynomials of degree $3T+1$

We show the univalence of $T$-symmetric Suffridge type polynomials $S_4^{(T)}$ in the unit disk, confirming thereby the conjecture proposed by Dmitrishin, Gray, and Stokolos in their recent paper. The result also implies the quasi-extremality of $S_n^{(T)}$ in the sense of Ruscheweyh.

math.CV↗
arXiv2021

Lattice points on small arcs

We show that for any $α\in (1/2,1)$ the number of lattice points belonging to an arc of length $R^α$ of the circle of radius $R$ centered at the origin is not uniformly bounded in $R$, which disproves the corresponding conjecture of Cilleruelo and Granville. We also give certain generalizations of this fact and estimates for the $L_4$-norm of Gauss sums.

math.NT↗
arXiv2021

John-Nirenberg inequality for Riemann type series

We obtain an improvement of the John-Nirenberg inequality for the series of the form $\sum_{n=1}^{\infty}n^{-1}e^{2πi n^k x},\;k>2,$ on intervals consisting of points of a same convergent of their continued fractions. We also establish a convergence criterion for these series.

math.CA↗
arXiv2020

Uniform convergence criterion for non-harmonic sine series

We show that for a nonnegative monotone sequence $\{c_k\}$ the condition $c_kk\to 0$ is sufficient for uniform convergence of the series $\sum_{k=1}^{\infty}c_k\sin k^α x$ on any bounded set for $α\in (0,2)$, and for an odd natural $α$ it is sufficient for uniform convergence on the whole $\mathbb{R}$. Moreover, the latter assertion still holds if we replace $k^α$ by any polynomial in odd powers with rational coefficients. On the other hand, in the case of an even $α$ it is necessary that $\sum_{k=1}^{\infty}c_k<\infty$ for convergence of the mentioned series at the point $π/2$ or at the point $2π/3$. Consequently, we obtain uniform convergence criteria. Besides, the results for a natural $α$ remain true for sequences from more general RBVS class.

math.CA↗