Points of middle density in the real line. (English) Zbl 1276.28009

Csörnyei, Marianna; Grahl, Jack; O’Neil, Toby C.

Real Anal. Exch. 37(2011-2012), No. 2, 243-248 (2012).

Let \(E\subset{\mathbb R}\) be a measurable set for the usual Lebesgue measure \(m\). For \(x\in{\mathbb R}\), let \(d_{E}(x,r):=\frac{1}{2r}m(E\cap\{y:|x-y|<r\})\), \(\underline{d}_{E}(x):=\liminf\limits_{r\downarrow 0}d_{E}(x,r)\) and \(\overline{d}_{E}(x):=\limsup\limits_{r\downarrow 0}d_{E}(x,r)\). A non-trivial Lebesgue measurable set \(E\) is called \(\delta\)-exceptional if, for any \(x\in{\mathbb R}\), we have either \(\underline{d}_{E}(x)<\delta\) or \(\overline{d}_{E}(x)>1-\delta\). Kolyada’s problem is to find the infimum \(\delta_0\) of those \(\delta\) for which a \(\delta\)-exceptional set exists.
A. Szenes [Adv. Math. 226, No. 1, 764–778 (2011; Zbl 1205.28001)] proved that \(\delta_{0}\leq 0.272\), where the upper bound is a positive root of the polynomial \(8x^3 +4x^2 +2x-1\), and conjectures that \(\delta_0\) equals to this root.
The authors of the present paper prove that \(\delta_0\) is smaller than the positive root \(0.2710\dots\) of the polynomial \(2x^3 +2x^2 +3x -1\), i.e., the Szenes conjecture is false.

Reviewer: Boris A. Kats (Kazan)

MSC:

28A75

Length, area, volume, other geometric measure theory

Keywords:

Lebesgue upper density; Lebesgue lower density

Citations:

Zbl 1205.28001

References:

[1]	V. I. Kolyada, On the metric Darboux property , Analysis Math., 9(4) (1983) 291-312 (in Russian). · Zbl 0544.26002 · doi:10.1007/BF01910308
[2]	A. Szenes, Exceptional points for Lebesgue’s density theorem in the real line , Advances in Math. 226 (1), (2011) 764-778. (http://arxiv.org/abs/math/0702432.) · Zbl 1205.28001 · doi:10.1016/j.aim.2010.07.011