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

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.

### MSC:

 28A75 Length, area, volume, other geometric measure theory

### Keywords:

Lebesgue upper density; Lebesgue lower density

Zbl 1205.28001
Full Text:

### 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
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