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 $\mathrm{Hst}(X)=T$, and apply this criterion to concrete existence and non-existence examples.