arXiv subjects

Paul Potgieter

Publications and source records attributed to Paul Potgieter.

6 recordsLinked to original sources

arXiv2021

Arithmetic properties of sparse subsets of $\mathbb{Z}^n$

Arithmetic progressions of length $3$ may be found in compact subsets of the reals that satisfy certain Fourier -- as well as Hausdorff -- dimensional requirements. It has been shown that a very similar result holds in the integers under analogous conditions, with Fourier dimension being replaced by the decay of a discrete Fourier transform. In this paper we make this correspondence more precise, using a well-known construction by Salem. Specifically, we show that a subset of the integers can be mapped to a compact subset of the continuum in a way which preserves certain dimensional properties as well as arithmetic progressions of arbitrary length. The higher-dimensional version of this construction is then used to show that certain parallelogram configurations must exist in sparse subsets of $\mathbb{Z}^n$ satisfying appropriate density and Fourier-decay conditions.

math.CA↗

arXiv2016

Algorithmically random Fourier series and Brownian motion

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 case of Fourier series on the circle with Rademacher coefficients. Finally, a specific Fourier series which has coefficients determined by a computable function is shown to converge to an algorithmically random Brownian motion.

math.PR↗

arXiv2015

The Fourier dimension of Brownian limsup fractals

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 original paper revealed several gaps in the arguments and a slight inaccuracy in the main theorem. This paper presents a new version of the construction and incorporates some recent results in order to establish a corrected version of Kaufman's theorem. The method of proof can then be extended to show that functionally determined rapid points of Brownian motion also form Salem sets for absolutely continuous functions of finite energy.

math.PR↗

arXiv2013

Nonstandard Hausdorff dimension in $\mathbb{R}^n$ and an application to the Kakeya conjecture in $\mathbb{R}^3$

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 a nonstandard proof of the Kakeya conjecture in two dimensions, which can then be extended to the three-dimensional case.

math.CA↗

arXiv2012

The rapid points of a complex oscillation

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 Brownian motion. Because of the nature of the proof we can then apply the concepts to so-called complex oscillations (or 'algorithmically random Brownian motion'), showing that their rapid points have the same dimension.

cs.CC↗

arXiv2010

Arithmetic progressions in Salem-type subsets of the integers

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 dimension of a compact subset of the reals. We then show that a version of a theorem of Łaba and Pramanik on 3-term arithmetic progressions in subsets of the unit interval also holds for subsets of the integers with fractional density and satisfying certain Fourier-decay conditions.

math.NT↗