We consider a planar convex body C and we prove several analogs of Roth's theorem on irregularities of distribution. When the boundary of C is C_2 we prove that for every set of N points in the unit square we have the sharp lower bound N^{1/4} for the L^2 discrepancy. When the boundary is only piecewise C_2 and is not a polygon we have the sharp lower bound N^{1/5}. We also give a whole range of intermediate sharp results between N^{1/5} and N^{1/4}. Our proofs depend on a lemma of Cassels-Montgomery, on ad hoc constructions of finite point sets, and on a geometric type estimate for the average decay of the Fourier transform of the characteristic function of C.

Brandolini, L., Travaglini, G. (2022). Irregularities of distribution and geometry of planar convex sets. ADVANCES IN MATHEMATICS, 396(12 February 2022) [10.1016/j.aim.2021.108162].

Irregularities of distribution and geometry of planar convex sets

Travaglini, Giancarlo
2022

Abstract

We consider a planar convex body C and we prove several analogs of Roth's theorem on irregularities of distribution. When the boundary of C is C_2 we prove that for every set of N points in the unit square we have the sharp lower bound N^{1/4} for the L^2 discrepancy. When the boundary is only piecewise C_2 and is not a polygon we have the sharp lower bound N^{1/5}. We also give a whole range of intermediate sharp results between N^{1/5} and N^{1/4}. Our proofs depend on a lemma of Cassels-Montgomery, on ad hoc constructions of finite point sets, and on a geometric type estimate for the average decay of the Fourier transform of the characteristic function of C.
Articolo in rivista - Articolo scientifico
Cassels-Montgomery lemma; Fourier transforms; Geometric discrepancy; Inner disk condition; Irregularities of distribution; Roth's theorem;
English
Brandolini, L., Travaglini, G. (2022). Irregularities of distribution and geometry of planar convex sets. ADVANCES IN MATHEMATICS, 396(12 February 2022) [10.1016/j.aim.2021.108162].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/343230
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