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Discrepancy estimates for some linear generalized monomials

Roswitha Hofer, Olivier Ramaré (2016)

Acta Arithmetica

We consider sequences modulo one that are generated using a generalized polynomial over the real numbers. Such polynomials may also involve the integer part operation [·] additionally to addition and multiplication. A well studied example is the (nα) sequence defined by the monomial αx. Their most basic sister, ( [ n α ] β ) n ≥ 0 , is less investigated. So far only the uniform distribution modulo one of these sequences is resolved. Completely new, however, are the discrepancy results proved in this paper. We show...

L p -discrepancy and statistical independence of sequences

Peter J. Grabner, Oto Strauch, Robert Franz Tichy (1999)

Czechoslovak Mathematical Journal

We characterize statistical independence of sequences by the L p -discrepancy and the Wiener L p -discrepancy. Furthermore, we find asymptotic information on the distribution of the L 2 -discrepancy of sequences.

On a problem of Sidon for polynomials over finite fields

Wentang Kuo, Shuntaro Yamagishi (2016)

Acta Arithmetica

Let ω be a sequence of positive integers. Given a positive integer n, we define rₙ(ω) = |(a,b) ∈ ℕ × ℕ : a,b ∈ ω, a+b = n, 0 < a < b|. S. Sidon conjectured that there exists a sequence ω such that rₙ(ω) > 0 for all n sufficiently large and, for all ϵ > 0, l i m n → ∞ r ₙ ( ω ) / n ϵ = 0 . P. Erdős proved this conjecture by showing the existence of a sequence ω of positive integers such that log n ≪ rₙ(ω) ≪ log n. In this paper, we prove an analogue of this conjecture in q [ T ] , where q is a finite field of q elements....

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