Constant Riesz potentials on a circle in a plane with an application to polarization optimality problems

Abstract

ABSTRACT:A characterization for a Riesz s-potential function of a multiset ω N of N points in 2 is given when s = 2−2N and the potential function is constant on a circle in 2 . The characterization allows us to partially prove a conjecture of Nikolov and Rafailov that if the potential function is constant on a circle Γ then the points in ω N should be equally spaced on a circle concentric to Γ . As an application of constant Riesz s-potential functions, we also find all maximal and minimal polarization constants and configurations of two concentric circles in 2 for certain values of s. KEYWORDS: roots of unity, max-min and min-max problems MSC2010: 52A40

Polarization, sign sequences and isotropic vector systems

“…In general, as n → ∞, the Taylor expansion of the cotangent gives where the second equality follows from a series computation described in Proposition 2.3 of [15]. We conclude the paper by restating the following natural conjecture of Bosuwan and Ruengrot [9], which is also supported by our numerical experiments:…”

“…They conjectured that the condition holding solely for p = 2n − 2 is already sufficient for characterization. This was verified by Bosuwan and Ruengrot [9] (for the case ω n ⊂ T , which we assumed anyway). The authors also proved that for p = 2, 4, .…”

“…This, along with (7), (8), (9), and the fact M p n M p (ω * n ), implies that M p n ∼ n µ p = n p p/2 as n → ∞. Proposition 5 shows that for integer exponents p with 0 < p < 2n, M p n = M p (ω * n ).…”

Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere

“…For potential functions bounded above, some authors also dealt with the problem of minimizing the maximum value of the potential, see [27,13,3]. We show here the optimality for this problem of a certain class of sharp spherical configurations.…”

“…The classical discrete polarization problem requires finding positions of N points on S d with the largest absolute minimum over S d of their total potential. It was studied in works [27,1,2,19,20,10,29,13,11,3,9]. In particular, for the Riesz and logarithmic potential on the sphere, the complete solution to the discrete polarization problem is known only for N points on the unit circle S 1 and for up to d + 2 points on S d , d ≥ 2.…”