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Contemporary Problems in Number Theory
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Limit distributions of normalized discrepancies of Kronecker sequences Bassam Fayad |
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Abstract: We study the Kronecker sequence There are two possible generalizations of the problem to higher dimensions : by either considering visits to balls, or to cubes. The behavior of the discrepancy is related to the so called small denominators of the corresponding frequency, and the difficulty in higher dimensions arises from the absence of a continued fraction theory with powerful metrical results as in the one dimension case. We rather study both situations, balls and cubes, using the ergodic theory of flows on homogeneous spaces. In the case of balls, we show that the discrepancy normalized by |