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Low-discrepancy point sets for non-uniform measures

Christoph Aistleitner, Josef Dick (2014)

Acta Arithmetica

We prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure μ on the d-dimensional unit cube. We improve a theorem of Beck, by showing that for any d ≥ 1, N ≥ 1, and any non-negative, normalized Borel measure μ on [ 0 , 1 ] d there exists a point set x 1 , . . . , x N ∈ [ 0 , 1 ] d whose star-discrepancy with respect to μ is of order D N * ( x 1 , . . . , x N ; μ ) ≪ ( ( l o g N ) ( 3 d + 1 ) / 2 ) / N . For the proof we use a theorem of Banaszczyk concerning the balancing of vectors, which implies an upper bound for the linear discrepancy...

Measures of maximal entropy for random β -expansions

Karma Dajani, Martijn de Vries (2005)

Journal of the European Mathematical Society

Let β > 1 be a non-integer. We consider β -expansions of the form ∑ i = 1 ∞ d i / β i , where the digits ( d i ) i ≥ 1 are generated by means of a Borel map K β defined on { 0 , 1 } ℕ × [ 0 , ⌊ β ⌋ / ( β − 1 ) ] . We show that K β has a unique mixing measure ν β of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure ν β the digits ( d i ) i ≥ 1 form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of β -expansions....

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