Towards BAD conjecture

Abstract

For $\alpha ,\beta ,\delta \in [0,1],\alpha +\beta =1$ we consider sets $${\mathrm{BAD}}^{\ast }(\alpha ,\beta ;\delta )=\left\{\xi =({\xi }_1,{\xi }_2)\in [0,1{]}^2:,\underset{p\in \mathbb{N}}{\inf }\max \{(p\log (p+1){)}^{\alpha }||p{\xi }_1||,(p\log (p+1){)}^{\beta }||p{\xi }_2||\}\ge \delta \right\}.$$ 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 $${\mathrm{BAD}}^{\ast }({\alpha }_1,{\beta }_1;\delta )\bigcap {\mathrm{BAD}}^{\ast }({\alpha }_2,{\beta }_2;\delta )\ne \varnothing .$$ Our result is based on A. Khintchine's construction and an original method due to Y. Peres and W. Schlag.

Minor correction of errors in Lemma 2