TY - GEN

T1 - The L2 Discrepancy of Two-Dimensional Lattices

AU - Bilyk, Dmitriy

AU - Temlyakov, Vladimir N.

AU - Yu, Rui

PY - 2013/1/1

Y1 - 2013/1/1

N2 - Let α be an irrational number with bounded partial quotients of the continued fraction ak. It is well known that symmetrizations of the irrational lattice have optimal order of L2 discrepancy, √log N. The same is true for their rational approximations, where pn/qn is the nth convergent of α. However, the question whether and when the symmetrization is really necessary remained wide open. We show that the L2 discrepancy of the nonsymmetrized lattice Ln(α) grows as in particular, characterizing the lattices for which the L2 discrepancy is optimal.

AB - Let α be an irrational number with bounded partial quotients of the continued fraction ak. It is well known that symmetrizations of the irrational lattice have optimal order of L2 discrepancy, √log N. The same is true for their rational approximations, where pn/qn is the nth convergent of α. However, the question whether and when the symmetrization is really necessary remained wide open. We show that the L2 discrepancy of the nonsymmetrized lattice Ln(α) grows as in particular, characterizing the lattices for which the L2 discrepancy is optimal.

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U2 - 10.1007/978-1-4614-4565-4_9

DO - 10.1007/978-1-4614-4565-4_9

M3 - Conference contribution

AN - SCOPUS:84883389739

SN - 9781461445647

T3 - Springer Proceedings in Mathematics and Statistics

SP - 63

EP - 77

BT - Recent Advances in Harmonic Analysis and Applications

PB - Springer New York LLC

ER -