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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, , 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...
We characterize statistical independence of sequences by the -discrepancy and the Wiener -discrepancy. Furthermore, we find asymptotic information on the distribution of the -discrepancy of sequences.
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,
.
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 , where is a finite field of q elements....
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