TY - GEN

T1 - The L2 Discrepancy of Irrational Lattices

AU - Bilyk, Dmitriy

PY - 2013

Y1 - 2013

N2 - It is well known that, when α has bounded partial quotients, the lattices{(k/N,{kα}) }N-1 k=0 have optimal extreme discrepancy. The situation with the L2 discrepancy, however, is more delicate. In 1956 Davenport established that a symmetrized version of this lattice has L2discrepancy of the orderf p √logN, which is the lowest possible due to the celebrated result of Roth. However, it remained unclear whether this holds for the original lattices without anymodifications. It turns out that the L2discrepancy of the lattice depends on much finer Diophantine properties of α, namely, the alternating sums of the partial quotients. In this paper we extend the prior work to arbitrary values of α and N. We heavily rely on Beck's study of the behavior of the sums Σ({kα}-1/2.

AB - It is well known that, when α has bounded partial quotients, the lattices{(k/N,{kα}) }N-1 k=0 have optimal extreme discrepancy. The situation with the L2 discrepancy, however, is more delicate. In 1956 Davenport established that a symmetrized version of this lattice has L2discrepancy of the orderf p √logN, which is the lowest possible due to the celebrated result of Roth. However, it remained unclear whether this holds for the original lattices without anymodifications. It turns out that the L2discrepancy of the lattice depends on much finer Diophantine properties of α, namely, the alternating sums of the partial quotients. In this paper we extend the prior work to arbitrary values of α and N. We heavily rely on Beck's study of the behavior of the sums Σ({kα}-1/2.

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U2 - 10.1007/978-3-642-41095-6_11

DO - 10.1007/978-3-642-41095-6_11

M3 - Conference contribution

AN - SCOPUS:84893444864

SN - 9783642410949

T3 - Springer Proceedings in Mathematics and Statistics

SP - 289

EP - 296

BT - Monte Carlo and Quasi-Monte Carlo Methods 2012

T2 - 10th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2012

Y2 - 13 February 2012 through 17 February 2012

ER -