On the Constructive Dimension Spectrum of Polynomials

TL;DR

This paper investigates the effective Hausdorff dimension spectra of polynomial curves, proving they contain at least two points and constructing polynomials with spectra of width greater than one, thus advancing understanding in fractal geometry.

Contribution

It establishes that polynomial curves have non-trivial dimension spectra, answers open questions about their properties, and constructs polynomials with spectra of width greater than one.

Findings

Dimension spectra of every polynomial curve contain at least two points.

Constructed polynomials with width of their spectra strictly greater than 1.

Resolved conjectures about the dimension spectra of polynomial curves and lines.

Abstract

Recently, Stull [18], [17] resolved a long-standing open problem posed by Lutz, on whether the set of effective Hausdorff dimensions of points on a straight line in $\mathbb{R}^2$ -- the effective dimension spectrum of the line -- contains a unit interval. This question is related to problems in classical fractal geometry like the Kakeya conjecture and Furstenberg sets. Stull posed an open question on the dimension spectra of polynomial curves. For the first result, with new techniques which adapt the theory of classical real root-finding of polynomials to the current setting, we show that the dimension spectra of every polynomial curve contains at least two points. This answers an open question posed by Stull [18], [17]. We use the main result to construct a class of polynomials which have width strictly greater than 1, answering a second problem stated in [18],[17]. Stull [18] resolved the dimension spectrum conjecture for planar lines, showing that it contains a unit interval. For the second result, we resolve the conjecture for a subfamily of polynomials whose coefficients form a "low" dimension point in $\mathbb{R}^{d+1}$.