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Ryutaro Misawa

Publications and source records attributed to Ryutaro Misawa.

5 recordsLinked to original sources
arXiv2026

Spherical Designs with Infinite Harmonic Strength

In this paper, we study the existence problem for spherical \(T\)-designs on the \(d\)-dimensional sphere, where \(T\) is an infinite subset of \(\mathbb N\). We show that, if \(d\ge 2\), then a finite subset of \(S^d\) has infinite harmonic strength if and only if it is antipodal. For \(d=1\), we show that infinite strength spherical designs are exactly cyclotomic designs, and we characterize their existence in terms of certain \(0\)-\(1\) polynomials. We also prove that the harmonic strength of every infinite strength spherical design has the weak GCD property. Finally, for a given infinite subset \(T\subset \mathbb N\) with the weak GCD property, we give a finite procedure to decide whether there exists \(X\subset S^1\) such that \(\operatorname{Hst}(X)=T\), and apply this criterion to concrete existence and non-existence examples.

math.CO↗
arXiv2026

Explicit construction of spherical $5$- and $7$-designs

This paper develops an explicit and implementable framework for constructing spherical designs by lifting point sets from tight fusion frames. By combining existing ingredients, we obtain, in every dimension, explicit spherical $5$-designs with $|X|=\mathcal{O}(d^3)$. As a core component of the method, we give an explicit construction of simplex $3$-designs realized as orbits of the symmetric group. Using these simplex designs as input, we further construct spherical $7$-designs in arbitrary even dimensions; more precisely, for every even integer $d\ge 6$ we obtain spherical $7$-designs in dimension $d$, and if $\frac{d}{2}-1$ is a prime power then the number of points is $\mathcal{O}(d^6)$.

math.CO↗
arXiv2026

Constructing spherical designs using tight $t$-fusion frames

In this paper, we study conditions under which a finite subset $Z$ of the unit sphere $S^{d-1}\subset \mathbb{R}^{d}$ becomes a spherical $t$-design, when $Z$ is constructed by the following procedure: starting from a finite set of $k$-dimensional subspaces in the real Grassmannian $G_{k,d}$, we place, for each such $k$-dimensional subspace, a finite set on its unit sphere, and then take the union of these sets in $S^{d-1}$. For this construction problem -- namely, obtaining spherical designs in higher dimensions by distributing point sets on lower-dimensional spheres subspace by subspace -- we provide a sufficient condition based on the framework of tight $t$-fusion frames ($\mathrm{TFF}_t$) due to Bachoc--Ehler. As a preparation for applications, we moreover give an explicit construction of equal-weight tight $2$-fusion frames on $G_{2,d}$ for infinitely many dimensions $d$, via unions of orbits of the hyperoctahedral group. We also derive necessary conditions for the existence of highly symmetric tight $t$-fusion frames, namely equi-chordal and equi-isoclinic tight $t$-fusion frames ($\mathrm{ECTFF}_t$ and $\mathrm{EITFF}_t$), on $G_{2,d}$, and in particular obtain bounds on the number of points.

math.CO↗
arXiv2025

Spherical Designs on $S^1$ of Finite Harmonic Strength

We study exact harmonic strengths of finite spherical designs on the unit circle. For a nonempty finite set \(X\subset S^1\), let \(\Hst(X)\) be the set of positive integers \(k\) for which the \(k\)-th complex moment \(P_k(X)=\sum_{x\in X}x^k\) vanishes. Equivalently, \(X\) is a spherical \(T\)-design precisely when \(T\subset \Hst(X)\). We consider the exact realization problem: given a finite set \(T\subset\mathbb N\), determine whether there exists a finite set \(X\subset S^1\) such that \(\Hst(X)=T\). We prove that every finite \(T\subset\mathbb N\) is realizable. More precisely, for each \(t\ge 1\) we construct uncountably many five-point sets with \(\Hst(X)=\{t\}\), and we prove that no smaller set can have this exact harmonic strength. A product construction then gives, for every finite \(T\subset\mathbb N\), a realization with \(|X|=5^{|T|}\). We also initiate the associated minimum-size problem \(N(T,2)\). We prove \(N(\{t\},2)=5\) for all \(t\ge1\), determine \(N(\{2,3\},2)=5\), and show that the optimal \(\{2,3\}\)-example is unique up to rotation. Finally, we discuss a rigid seven-point example related to \(T=\{2,3,4,10\}\).

math.CO↗