arXiv subjects

Danylo Radchenko

Publications and source records attributed to Danylo Radchenko.

At least 19 recordsLinked to original sources

arXiv2026

Real quadratic fields and finite quantum dilogarithms I

We prove that Stark-Shintani ray class invariants (Stark units) associated to real quadratic fields are algebraic numbers. These invariants are given by special values of Faddeev's modular quantum dilogarithm, introduced by Garoufalidis-Kashaev-Zagier. Our main discovery is that special values of modular quantum dilogarithm satisfy an explicit overdetermined system of polynomial equations, matching a variation on the defining equations of Andersen-Kashaev's notion of a quantum dilogarithm on a product of two cyclic groups. Solutions to this system of equations can be used to categorify fusion rings introduced by Izumi, and the algebraicity of the special values then follows by Ocneanu's rigidity theorem. As a byproduct, we obtain an explicit infinite family of irrational near-group fusion categories. As a further application, we prove a family of quadratic relations for Stark units recently conjectured by Appleby, Flammia, and Kopp motivated by Zauner's conjecture about SIC-POVMs (complex equiangular lines).

math.NT↗

arXiv2026

The $D^6 R^4$ interaction as a Poincaré series, and a related shifted convolution sum

We complete the program, initiated in a 2015 paper of Green, Miller, and Vanhove, of directly constructing the automorphic solution to the string theory $D^6 R^4$ differential equation $(Δ-12)f=-E_{3/2}^2$ for $SL(2,\Z)$. The construction is via a type of Poincaré series, and requires explicitly evaluating a particular double integral. We also show how to use double Dirichlet series to formally derive the predicted vanishing of one type of term appearing in $f$'s Fourier expansion, confirming a conjecture made by Chester, Green, Pufu, Wang, and Wen motivated by Yang-Mills theory (and later proved rigorously by Fedosova, Klinger-Logan, and Radchenko using the Gross-Zagier Holomorphic Projection Lemma.).

math.NT↗

arXiv2026

The sphere packing problem in dimension 24

Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.

math.NT↗

arXiv2026

The Hörmander--Bernhardsson function in higher dimensions

We study the problem of finding the norm of the point evaluation operator in the Paley--Wiener space $PW^{1}(\r^d)$, consisting of $d$-variable functions of spherical exponential type that are integrable on $\r^d$. The extremal functions can be taken radial, which naturally leads us to consider a related extremal problem in a weighted Paley--Wiener space of single-variable functions. We establish that the radial extremal function must satisfy a third-order linear ODE with polynomial coefficients for every $d \geq 1$, extending Gorbachev's recent odd-dimensional result. Along the way, we prove interpolation and reciprocal formulas involving the zeros of the extremizer.

math.CA↗

arXiv2026

On Grünbaum's problem for symmetric configurations

Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as a finite-alphabet squared-error rate-distortion supremum $α_0$. Specialized to the two-point case, i.e., for subsets of Boolean cubes, this gives the explicit lower bound \[g_n\ge (1.160235457\ldots-o(1))^n,\] improving the previous best bound $(2/\sqrt3-o(1))^n$. Using Fix's Gaussian characterization of the rate-distortion problem, we give a numerical three-point construction with exponent base greater than $1.160497831$. Finally, we show that $α_0$ is not attained by any finitely supported distribution.

math.MG↗

arXiv2026

Sharp Lower Bounds for Sumsets in Hypercubes

We prove a sharp lower bound for the cardinality of sumsets of subsets of $\mathbb{Z}^d$ confined to a hypercube, resolving in strong form a conjecture that was made explicit by Becker, Ivanisvili, Krachun and Madrid and had circulated in the folklore of the field for some time. Specifically, for sets $A_j\subseteq \{0,1,2,\dots,m\}^d$ we show that \[|A_1+\dots+A_n|\;\geq\; (|A_1|\cdots|A_n|)^{1/p},\qquad p=\frac{n\log(m+1)}{\log(nm+1)},\] with the exponent best possible. The only previously known sharp cases were $A_j\subseteq \{0,1\}^d$, for all $n\ge1$, and $A_j\subseteq \{0,1,2\}^d$ for $n=2$. We also prove a sharp inequality in the case when $A_j\subseteq\{0,1,\dots,m_j\}^d$ for different $m_j$. We obtain the above inequality as a corollary of a stronger result on sup-convolution of functions on $\mathbb{Z}^d$, whose proof is based on a novel mixed volume representation of a lattice path norm, together with a sharp one-dimensional functional inequality.

math.CO↗

arXiv2026

A construction of spherical $5$-designs with $O(d^2)$ points

For every $d\geq1$ we give an explicit equal-weight spherical $5$-design in $\mathbb{S}^{d-1}\subset\mathbb{R}^d$ with at most $72d^2$ points. Our approach utilizes recent construction of complex projective $2$-designs based on Sidon sets.

math.CO↗

arXiv2026

An integrality phenomenon

We prove a general statement about the integrality of the sequences generated by a recursion of the following form: $nu_n$ equals a linear combination of $u_{n-1},u_{n-2},\dots,u_0$ with polynomial coefficients in $n$ of special form. This includes a conjectural integrality of the sequence related to the Hörmander-Bernhardsson extremal function, for which we further give a direct proof as well.

math.NT↗

arXiv2026

Multiple polylogarithms and the Steinberg module

We establish a connection between multiple polylogarithms on a torus and the Steinberg module of $\mathbb{Q}$, and show that multiple polylogarithms of depth $d$ and weight $n$ can be expressed via a single function $\mathrm{Li}_{n-d+1,1,\dots,1}(x_1,x_2,\dots,x_d)$. Using this connection, we give a simple proof of the Bykovski\uı theorem, explain the duality between multiple polylogarithms and iterated integrals, and provide a polylogarithmic interpretation of the conjectures of Rognes and Church-Farb-Putman.

math.NT↗

arXiv2026

The Hörmander--Bernhardsson extremal function

We characterize the function $φ$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $π$ for which $f(0)=1$. This function, studied by Hörmander and Bernhardsson in 1993, has only real zeros $\pm τ_n$, $n=1,2, \ldots$. Starting from the fact that $n+\frac12-τ_n$ is an $\ell^2$ sequence, established in an earlier paper of ours, we identify $φ$ in the following way. We factor $φ(z)$ as $Φ(z)Φ(-z)$, where $Φ(z)= \prod_{n=1}^\infty(1+(-1)^n\frac{z}{τ_n})$ and show that $Φ$ satisfies a certain second order linear differential equation along with a functional equation either of which characterizes $Φ$. We use these facts to establish an odd power series expansion of $n+\frac12-τ_n$ in terms of $(n+\frac12)^{-1}$ and a power series expansion of the Fourier transform of $φ$, as suggested by the numerical work of Hörmander and Bernhardsson. The dual characterization of $Φ$ arises from a commutation relation that holds more generally for a two-parameter family of differential operators, a fact that is used to perform high precision numerical computations.

math.CA↗

arXiv2025

The basis functions of Fourier interpolation

The basis functions of the Fourier interpolation formula of Radchenko and Viazovska, constructed by means of weakly holomorphic modular forms for the Hecke theta group, are entire functions of order $2$ having interesting time-frequency properties. We give precise size estimates and study the distribution of zeros of these functions. We give in particular asymptotic estimates for the location and the number of extraneous zeros on or close to the real line. This result reveals the surprising existence of Fourier nonuniqueness pairs whose apparent ``excess'' compared to the Fourier uniqueness pair of Radchenko and Viazovska may be made arbitrarily large. Our estimates also show that the basis functions fail to yield a Riesz basis in the Hilbert space used by Kulikov, Nazarov, and Sodin in their recent study of Fourier uniqueness pairs. Some numerical data are presented, suggesting additional fine scale properties.

math.NT↗

arXiv2025

Small Volume Bodies of Constant Width with Tetrahedral Symmetries

For every $n\ge 2$, we construct a body $U_n$ of constant width $2$ in $\mathbb{E}^n$ with small volume and symmetries of a regular $n$-simplex. $U_2$ is the Reuleaux triangle. To the best of our knowledge, $U_3$ was not previously constructed, and its volume is smaller than the volume of other three-dimensional bodies of constant width with tetrahedral symmetries. While the volume of $U_3$ is slightly larger than the volume of Meissner's bodies of width $2$, it exceeds the latter by less than $0.137\%$. For all large $n$, the volume of $U_n$ is smaller than the volume of the ball of radius $0.891$.

math.MG↗

arXiv2025

The Hörmander--Bernhardsson extremal function: A preliminary study

We study the function $φ_1$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $π$ for which $f(0)=1$. This function, first studied by Hörmander and Bernhardsson in 1993, has only real zeros $\pm τ_n$, $n=1,2, \ldots$, and the sequence $(τ_n-n-\frac12)$ has $\ell^2$ norm bounded by $0.13$. The zeros $τ_n$ can be computed by means of a fixed point iteration.

math.FA↗

arXiv2025

On asymptotic Lebesgue's universal covering problem

Universal cover in $\mathbb{E}^{n}$ is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for the convex set of smallest area that serves as a universal cover in the plane ($n=2$). A simple universal cover in $\mathbb{E}^n$ is provided by the classical theorem of Jung, which states that any set of diameter 1 in an $n$-dimensional Euclidean space is contained in a ball $J_n$ of radius $\sqrt{\tfrac{n}{2n+2}}$; in other words, $J_n$ is a universal cover in $\mathbb{E}^n$. We show that in high dimensions, Jung's ball $J_n$ is asymptotically optimal with respect to the volume, namely, for any universal cover $U \subset \mathbb{E}^n$, $$ {\rm Vol}(U) \ge (1-o(1))^n{\rm Vol}(J_n). $$

math.MG↗

arXiv2025

Fourier interpolation in dimensions 3 and 4 and real-variable Kloosterman sums

We give a construction of radial Fourier interpolation formulas in dimensions 3 and 4 using Maass--Poincaré type series. As a corollary we obtain explicit formulas for the basis functions of these interpolation formulas in terms of what we call real-variable Kloosterman sums, which were previously introduced by Stoller. We also improve the bounds on the corresponding basis functions $a_{n,d}(x)$, $d=3,4$, for fixed $x$, in terms of the index $n$.

math.NT↗

arXiv2025

The Kirkwood-Dirac representation associated to the Fourier transform for finite abelian groups: positivity

We construct and study the Kirkwood-Dirac (KD) representations naturally associated to the Fourier transform of finite abelian groups $G$. We identify all pure KD-positive states and all KD-real observables for these KD representations. We provide a necessary and sufficient condition ensuring that all KD-positive states are convex combinations of pure KD-positive states. We prove that for $G=\Z_{d}$, with $d$ a prime power, this condition is satisfied. We provide examples of abelian groups where it is not. In those cases, the convex set of KD-positive states contains states outside the convex hull of the pure KD-positive states.

quant-ph↗

arXiv2025

The Hopf algebra of formal multiple polylogarithms

We define a Hopf algebra of polylogarithms of an arbitrary field, which is a candidate for a conjectural Hopf algebra of framed mixed Tate motives. Our definition is elementary and mimics Goncharov's construction of higher Bloch groups. We also discuss the Hodge and motivic realizations of the Hopf algebra of polylogarithms.

math.NT↗

arXiv2025

Small volume bodies of constant width

For every large enough $n$, we explicitly construct a body of constant width $2$ that has volume less than $0.9^n \text{Vol}(\mathbb{B}^{n}$), where $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$. This answers a question of O.~Schramm.

math.MG↗