Towards BAD conjecture

TL;DR

This paper investigates the intersection properties of certain sets related to Diophantine approximation, proving non-empty intersections under specific conditions using advanced mathematical constructions.

Contribution

It establishes the non-empty intersection of BAD sets with different parameters, extending previous results with a novel method based on Khintchine's construction and Peres-Schlag techniques.

Findings

Proves intersections of BAD sets are non-empty for certain parameters.

Extends understanding of Diophantine approximation sets.

Introduces a new method combining classical and modern techniques.

Abstract

For $\alpha, \beta, \delta \in [0,1], \alpha +\beta = 1 $ we consider sets $$ {\rm BAD}^* (\alpha, \beta ;\delta) = \left\{\xi = (\xi_1,\xi_2) \in [0,1]^2: ,\inf_{p\in \mathbb{N}} \max \{(p\log(p+1))^\alpha ||p\xi_1||, (p\log (p+1))^\beta ||p\xi_2||\} \ge \delta \right\}. $$ We prove that for different $(\alpha_1,\beta_1), (\alpha_2,\beta_2), \alpha_1 +\beta_1 = \alpha_2 +\beta_2 = 1 $ and $\delta $ small enough $$ {\rm BAD}^* (\alpha_1, \beta_1 ;\delta) \bigcap {\rm BAD}^* (\alpha_2, \beta_2 ;\delta) \neq \varnothing . $$ Our result is based on A. Khintchine's construction and an original method due to Y. Peres and W. Schlag.