7 papers
Algorithmically random Fourier series and Brownian motion
Paul Potgieter
We consider some random series parametrised by complex binary strings. The simplest case is that of Rademacher series, independent of a time parameter. This is then extended to the…
Arithmetic properties of sparse subsets of
Paul Potgieter
Arithmetic progressions of length may be found in compact subsets of the reals that satisfy certain Fourier -- as well as Hausdorff -- dimensional requirements. It has been sho…
Nonstandard Hausdorff dimension in and an application to the Kakeya conjecture in
Paul Potgieter
Through the use of a nonstandard version of Frostman's lemma, the notion of Hausdorff dimension is formulated in nonstandard euclidean space of arbitrary dimension. This allows for…
The Fourier dimension of Brownian limsup fractals
Paul Potgieter
Robert Kaufman's proof that the set of rapid points of Brownian motion has a Fourier dimension equal to its Hausdorff dimension was first published in 1974. A study of the proof of…
The rapid points of a complex oscillation
Paul Potgieter
By considering a counting-type argument on Brownian sample paths, we prove a result similar to that of Orey and Taylor on the exact Hausdorff dimension of the rapid points of Brown…
Arithmetic progressions in Salem-type subsets of the integers
Paul Potgieter
Given a subset of the integers of zero density, we define the weaker notion of fractional density of such a set. It is shown how this notion corresponds to that of the Hausdorff di…