2019/07/18 by Marcin Michalski, Michalski, Marcin
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Bijection #Combinatorics #Computer science #Construct (python library) #Discrete mathematics #Existential quantification #Geometry #Mathematics #Measure (data warehouse) #Optimization and Variational Analysis #Plane (geometry) #Point processes and geometric inequalities #Rational number #Set (abstract data type) #math.GN #math.MG
paper · pdf · doi:10.48550/arxiv.1907.09385
Conference paper: $13^{th}$ Students' Science Conference (2015), Polanica-Zdrój, Poland
arxiv created 2019/07/18 · openalex publication_date 2019/07/18 · arxiv updated 2019/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper we shall give a short proof of the result originally obtained by Ashutosh Kumar that for each A⊂ ℝ there exists B⊂ A full in A such that no distance between two distinct points from B is rational. We will construct a Bernstein subset of ℝ which also avoids rational distances. We will show some cases in which the former result may be extended to subsets of ℝ2, i. e. it remains true for measurable subsets of the plane and if non(N)=cof(N) then for a given set of positive outer measure we may find its full subset which is a partial bijection and avoids rational distances.