TY - JOUR

T1 - Single radius spherical cap discrepancy via gegenbadly approximable numbers

AU - Bilyk, Dmitriy

AU - Mastrianni, Michelle

AU - Steinerberger, Stefan

N1 - Publisher Copyright:
© 2024 Elsevier Inc.

PY - 2024/8

Y1 - 2024/8

N2 - A celebrated result of Beck shows that for any set of N points on Sd there always exists a spherical cap B⊂Sd such that number of points in the cap deviates from the expected value σ(B)⋅N by at least N1/2−1/2d, where σ is the normalized surface measure. We refine the result and show that, when d≢1(mod4), there exists a (small and very specific) set of real numbers such that for every r>0 from the set one is always guaranteed to find a spherical cap Cr with the given radius r for which the result holds. The main new ingredient is a generalization of the notion of badly approximable numbers to the setting of Gegenbauer polynomials: these are fixed numbers x∈(−1,1) such that the sequence of Gegenbauer polynomials (Cnλ(x))n=1∞ avoids being close to 0 in a precise quantitative sense.

AB - A celebrated result of Beck shows that for any set of N points on Sd there always exists a spherical cap B⊂Sd such that number of points in the cap deviates from the expected value σ(B)⋅N by at least N1/2−1/2d, where σ is the normalized surface measure. We refine the result and show that, when d≢1(mod4), there exists a (small and very specific) set of real numbers such that for every r>0 from the set one is always guaranteed to find a spherical cap Cr with the given radius r for which the result holds. The main new ingredient is a generalization of the notion of badly approximable numbers to the setting of Gegenbauer polynomials: these are fixed numbers x∈(−1,1) such that the sequence of Gegenbauer polynomials (Cnλ(x))n=1∞ avoids being close to 0 in a precise quantitative sense.

KW - Gegenbauer polynomials

KW - Spherical cap discrepancy

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U2 - 10.1016/j.aim.2024.109812

DO - 10.1016/j.aim.2024.109812

M3 - Article

AN - SCOPUS:85197080944

SN - 0001-8708

VL - 452

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 109812

ER -