Spherical Designs with Infinite Harmonic Strength

Abstract: 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.

• The paper establishes that for dimensions d ≥ 2, a finite design has infinite harmonic strength if and only if it is antipodal.
• It characterizes S¹ designs by proving that only cyclotomic designs achieve infinite harmonic strength through a rich periodic structure.
• Using algebraic techniques and polynomial criteria, the work provides explicit decidability conditions and bounds linking harmonic strength with combinatorial properties.

Spherical Designs with Infinite Harmonic Strength

Introduction and Framework

The study of spherical designs—finite subsets of the $d$-dimensional sphere $S^d$ whose pointwise averages match exact spherical averages for classes of polynomials—sits at the intersection of combinatorics, harmonic analysis, and algebraic geometry. For a non-empty finite set $X\subset S^d,$ the traditional notion of a spherical $t$-design requires that $$\frac{1}{|S^d|}\int_{S^d} f(\xi)d\xi = \frac{1}{|X|}\sum_{\xi\in X} f(\xi)$$ hold exactly for all polynomials $f$ of degree at most $t$. Extending this, the concept of spherical $T$-designs considers arbitrary (possibly infinite) sets of polynomial degrees $T\subset\mathbb{N}$, leading to the central notion of harmonic strength: $\operatorname{Hst}(X) = \left\{ k\in\mathbb{N} : \sum_{\xi\in X} P(\xi) = 0 \text{ for all } P\in\Harm_k(d+1)\right\}$ where $S^d$0 is the space of degree-$S^d$1 homogeneous harmonic polynomials in $S^d$2 variables.

The paper investigates the existence and classification of finite $S^d$3 with infinite $S^d$4 (termed infinite strength spherical designs), providing a sharp dichotomy based on the dimension $S^d$5 and uncovering strong algebraic constraints on the possible $S^d$6 for which such $S^d$7 can exist.

Classification of Infinite Strength Spherical Designs

Higher Dimensions: $S^d$8

For $S^d$9, the main result is a complete classification:

Theorem: A finite subset $X\subset S^d,$0 is an infinite strength spherical design if and only if $X\subset S^d,$1 is antipodal (i.e., $X\subset S^d,$2).

The proof leverages inequalities for Jacobi (specifically, Gegenbauer) polynomials [HS2014] applied to the duality of harmonic polynomials and pointwise structure on the sphere. Using these inequalities, the authors show that for non-antipodal $X\subset S^d,$3 and $X\subset S^d,$4, the harmonic strength can only contain finitely many indices—eventually, the moment sum condition fails except possibly for odd degrees if and only if $X\subset S^d,$5 maintains antipodal symmetry.

Moreover, the paper gives explicit (dimension and set-dependent) upper bounds on the maximal degree occurring in the harmonic strength, indicating, for arbitrary finite $X\subset S^d,$6 in $X\subset S^d,$7 with $X\subset S^d,$8, that only for small $X\subset S^d,$9 can $t$0 hold unless antipodality is present.

If $t$1 is antipodal, then all odd $t$2 belong to $t$3, and the even part is finite and explicitly bounded.

Dimension $t$4: Cyclotomic Designs

In the case $t$5, the landscape changes dramatically. The unit circle admits much richer combinatorial structures. Here the following classification is established:

Theorem: For $t$6, $t$7 has infinite harmonic strength if and only if $t$8 is a cyclotomic design.

Cyclotomic designs generalize both antipodal and group-type designs. The paper provides an algebraic characterization for $t$9: $$\frac{1}{|S^d|}\int_{S^d} f(\xi)d\xi = \frac{1}{|X|}\sum_{\xi\in X} f(\xi)$$0 is a cyclotomic design if its points can be partitioned into nontrivial sets each forming a regular polygon (possibly with distinct periods) modulated by a group of roots of unity.

Using complex moments—a scalar representation via $$\frac{1}{|S^d|}\int_{S^d} f(\xi)d\xi = \frac{1}{|X|}\sum_{\xi\in X} f(\xi)$$1—and leveraging the Skolem–Mahler–Lech theorem [Lech, Skolem] on the zeros of recurrence sequences, the authors show that $$\frac{1}{|S^d|}\int_{S^d} f(\xi)d\xi = \frac{1}{|X|}\sum_{\xi\in X} f(\xi)$$2 with infinite support must exhibit a periodic structure, corresponding precisely to the combinatorics of cyclotomic partitions.

Harmonic Strength and the Weak GCD Property

A central algebraic result is that the infinite subsets $$\frac{1}{|S^d|}\int_{S^d} f(\xi)d\xi = \frac{1}{|X|}\sum_{\xi\in X} f(\xi)$$3 which occur as harmonic strengths of finite designs must possess the weak GCD property: there exist finite $$\frac{1}{|S^d|}\int_{S^d} f(\xi)d\xi = \frac{1}{|X|}\sum_{\xi\in X} f(\xi)$$4, period $$\frac{1}{|S^d|}\int_{S^d} f(\xi)d\xi = \frac{1}{|X|}\sum_{\xi\in X} f(\xi)$$5, and $$\frac{1}{|S^d|}\int_{S^d} f(\xi)d\xi = \frac{1}{|X|}\sum_{\xi\in X} f(\xi)$$6 such that

$$\frac{1}{|S^d|}\int_{S^d} f(\xi)d\xi = \frac{1}{|X|}\sum_{\xi\in X} f(\xi)$$7

For $$\frac{1}{|S^d|}\int_{S^d} f(\xi)d\xi = \frac{1}{|X|}\sum_{\xi\in X} f(\xi)$$8, any antipodal $$\frac{1}{|S^d|}\int_{S^d} f(\xi)d\xi = \frac{1}{|X|}\sum_{\xi\in X} f(\xi)$$9 realizes only $f$0 with $f$1 (i.e., odd $f$2 plus a finite set), while for $f$3, cyclotomic designs realize those $f$4 with arbitrarily large periods, but no further.

A highly nontrivial aspect is proving that $f$5 for any infinite strength design must be periodic in this strong sense; the proof relies on transcendence results (Baker's theorem [B1975]) and careful analysis of the algebraic representation of $f$6 via polynomials.

Decidability: Existence for a Given $f$7

The inverse problem—given $f$8 with the weak GCD property, does some $f$9 exist with $t$0?—is resolved constructively.

Theorem: Such $t$1 exists if and only if there exists a finite collection of nonzero $t$2-$t$3 polynomials $t$4 of degree $t$5 such that

$t$6

where $t$7 is the $t$8-th cyclotomic polynomial.

This algebraic criterion provides a reduction to finite computation (as there are finitely many $t$9-$T$0 polynomials of degree $T$1), making the existence of infinite strength designs fully decidable for any candidate $T$2. The paper demonstrates both explicit constructions (e.g., for all $T$3 corresponding to $T$4) and nonexistence, such as for $T$5 missing all indices with certain residues modulo $T$6.

Bounds and Fisher-Type Inequalities

The paper compares its results with Fisher-type lower bounds on spherical designs [D1977]. For $T$7, the size of $T$8 with prescribed harmonic strength must grow at least polynomially in the degree, even for single $T$9, with bounds depending on the maximal inner product among points. These are sharp in various regimes and demonstrate that antipodal or cyclotomic structure does not lead to "too small" infinite strength sets for high degrees.

Implications and Future Directions

The results have deep implications for both the combinatorics of design theory and harmonic analysis. In higher dimensions, infinite strength is sharply limited to antipodal sets, highlighting robust rigidity. For $T\subset\mathbb{N}$0, the picture is richer: the algebraic machinery not only yields classification, but also algorithmic existential answers, and opens connections to the geometry of cyclotomic fields and the algebraic theory of moments.

The algebraic approach, grounded in sophisticated tools such as transcendence theory and recurrence sequence theorems, is striking in its scope and applicability. The methods suggest explicit strategies for constructing or ruling out spherical designs with intricate combinatorial properties, with potential implications for coding theory, numerical integration, and harmonic analysis on other spaces.

Open directions include the quantification and minimization of $T\subset\mathbb{N}$1 for prescribed $T\subset\mathbb{N}$2, further refinement of periodicity constraints, and generalization to more exotic manifolds or spaces with other symmetry groups. Optimization problems regarding the minimal size or configuration for a given harmonic strength are also left as challenging avenues for subsequent work.

Conclusion

This paper delivers a rigorous characterization of infinite strength spherical designs, demonstrating that in all dimensions but one, antipodality is necessary, and for $T\subset\mathbb{N}$3, designs must be cyclotomic. The mapping of harmonic strength to algebraic properties of polynomials and explicit decidability criteria fully resolves the existence question for infinite $T\subset\mathbb{N}$4-designs. The results integrate sharp combinatorial, analytic, and algebraic analysis, offering an authoritative reference for both the theory and constructive aspects of spherical $T\subset\mathbb{N}$5-designs.

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References:

• "Spherical Designs with Infinite Harmonic Strength" (2607.01761)
• Haagerup & Schlichtkrull, "Inequalities for Jacobi polynomials" [HS2014]
• Delsarte, Goethals, & Seidel, "Spherical codes and designs"
• Baker, "Transcendental Number Theory" [B1975]
• Lech, "A note on recurring series"; Skolem, "Einige Sätze über gewisse Reihenentwicklungen..."