arXiv subjects
Publications and source records attributed to Aljoša Volčič.
In this paper we will prove that for any planar measurable set of finite measure $M$, its successive Steiner symmetrizations with respect to the Kakutani-Fibonacci sequence of directions converge to the ball $M^*$ centered at the origin and having the same measure.
In this paper we study area-filling curves, i.e. continuous and injective mappings defined on [0,1] whose graph has positive measure. Current literature calls them ``Osgood curves", but their invention is due to H. Lebesgue. Stromberg and Tseng constructed homogeneous area-filling curves and offered an elegant example. We show that an appropriate variant of Knopp's construction attains the same homogeneity result. In Section 4 we discuss briefly the existence of an ``invasive" curve, i.e. a continuous and injective mapping from the half-open interval [0,1[ to the unit square, whose image has measure 1. In the last section we discuss several aspects of the Lance-Thomas curve, connecting it with the other construction due to Stromberg and Tseng.
We define a countable family of sequences of points in the unit square: the {\it $LS$-sequences of points à la Halton}. They reveal a very strange and interesting behaviour, as well as resonance phenomena, for which we have not found an explanation, so far. We conclude with three open problems.
In this paper we consider the sequence of Kakutani's $α$-refinements corresponding to the inverse of golden ratio (which we call Kakutani-Fibonacci sequence of partitions) and associate to it an ergodic interval exchange (which we call Kakutani-Fibonacci transformation) using the "cutting-stacking" technique. We prove that the orbit of the origin under this map coincides with a low discrepancy sequence (which we call Kakutani-Fibonacci sequence of points), which has been also considered by other authors.