Badly approximable vectors on a vertical Cantor set

2012/03/31 by Erez Nesharim, Nesharim, Erez · 1 citation
Mathematics · #Approximation Theory and Sequence Spaces #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.DS #math.NT

paper · pdf · doi:10.48550/arxiv.1204.0110

openalex publication_date 2012/03/31 · arxiv created 2013/07/09 · arxiv updated 2013/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For i, j > 0, i + j = 1, the set of badly approximable vectors with weight (i, j) is defined by Bad(i, j) = \(x, y) ∈ \R² : ∃ c > 0 ∀ q∈\N, max\q||qx||^1/i, q||qy||^1/j \ > c\, where ||x|| is the distance of x to the nearest integer. In 2010 Badziahin-Pollington-Velani solved Schmidt's conjecture which was stated in 1982, proving that Bad(i, j) ∩ Bad(j, i) is nonempty. Using Badziahin-Pollington-Velani's technique with reference to fractal sets, we were able to improve their results: Assume that we are given a sequence (i_t, j_t) with i_t, j_t > 0, i_t + j_t = 1. Then, the intersection of Bad(i_t, j_t) over all t is nonempty.

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