Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Center of distances and Bernstein sets

Mateusz Kula

TL;DR

This work resolves Filipczak's question by showing that for every $A\subset [0,\infty)$ with $0\in A$ there exists $X\subset R$ with center of distances $S(X)=A$, and moreover that $X$ can be chosen as a Bernstein set. The authors develop an average-operator framework $T_C$ and a transfinite, Hamel-basis–driven construction to achieve surjectivity of the center-of-distances operator, embedding the prescribed centers into the complement of the dyadic-averaging closure $T_C^\infty(Y)$. They then refine the construction to produce Bernstein sets realizing $S(X)=A$, including the full case $S(X)=[0,\infty)$. Overall, the results give a complete answer to the posed problem and expand techniques for fabricating sets with prescribed distance-centered structures on the real line.

Abstract

We show that for any subset $A\subset [0,\infty)$, where $0\in A$, there exists a Bernstein set $X\subset \mathbb R$ such that $A$ is the center of distances of $X$.

Center of distances and Bernstein sets

TL;DR

This work resolves Filipczak's question by showing that for every with there exists with center of distances , and moreover that can be chosen as a Bernstein set. The authors develop an average-operator framework and a transfinite, Hamel-basis–driven construction to achieve surjectivity of the center-of-distances operator, embedding the prescribed centers into the complement of the dyadic-averaging closure . They then refine the construction to produce Bernstein sets realizing , including the full case . Overall, the results give a complete answer to the posed problem and expand techniques for fabricating sets with prescribed distance-centered structures on the real line.

Abstract

We show that for any subset , where , there exists a Bernstein set such that is the center of distances of .

Paper Structure

This paper contains 4 sections, 4 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. On the average operator
  3. Surjectivity of the center of distances operator
  4. Bernstein sets and center of distances

Key Result

Lemma 1

Given $C\subset \mathbb R$, $Y\subset \mathbb R\setminus C$ and $v\in T^n_C(Y)$, there exists a function $d\colon 2^{\leqslant n}\rightarrow \mathbb R$ such that Moreover, if $Y=\{x-b,x+b\}$ and $x\in C$, then $d(s)=x-b$ for all $s\in 2^{\leqslant n}$ or $d(s)=x+b$ for all $s\in 2^{\leqslant n}$.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Proposition 3
  • proof
  • Theorem 4
  • proof