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On some properties of Erdős sets. (English) Zbl 0546.39001
Let H be a Hamel basis of $${\mathbb{R}}^ N$$. The set $$H^*$$ of all finite linear combinations of the elements of H with integral coefficients is called the Erdős set associated with H. Independently of what Hamel basis H we start, the set H always is saturated non-measurable [for the case $$N=1$$ see P. Erdős, ibid. 10, 267-269 (1963; Zbl 0123.320)]. The main result of the paper reads as follows. Let $$A\subset {\mathbb{R}}^ N$$ be an arbitrary Lebesgue measurable set of positive measure or a second category Baire set, let H be an arbitrary Hamel basis of $${\mathbb{R}}^ N$$, and put $$T=A\cap H^*$$. If $$D\subset {\mathbb{R}}^ N$$ is an open and convex set containing T, and f:$$D\to {\mathbb{R}}$$ is a (Jensen) convex function such that f is bounded above on T or the restriction $$f| T$$ is continuous, then f is continuous in D.

##### MSC:
 39B99 Functional equations and inequalities 39B72 Systems of functional equations and inequalities
Zbl 0123.320
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