Hausdorff dimension of weighted singular vectors in $\mathbb R^2$

Lingmin Liao; Ronggang Shi; Omri N. Solan; Nattalie Tamam

doi:10.4171/jems/934

JournalsjemsVol. 22, No. 3pp. 833–875

Hausdorff dimension of weighted singular vectors in $R^{2}$

Lingmin Liao
Université Paris-Est Créteil, France
Ronggang Shi
Fudan University, China
Omri N. Solan
Tel Aviv University, Israel
Nattalie Tamam
Tel Aviv University, Israel

$Hausdorff dimension of weighted singular vectors in $\mathbb R^2$ cover$

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Abstract

Let $w = (w_{1}, w_{2})$ be a pair of positive real numbers with $w_{1} + w_{2} = 1$ and $w_{1} \geq w_{2}$ . We show that the set of $w$ -weighted singular vectors in $R^{2}$ has Hausdorff dimension $2 - \frac{1}{1 + w _{1}}$ . This extends the previous work of Yitwah Cheung on the Hausdorff dimension of the usual (unweighted) singular vectors in $R^{2}$ .

Cite this article

Lingmin Liao, Ronggang Shi, Omri N. Solan, Nattalie Tamam, Hausdorff dimension of weighted singular vectors in $R^{2}$ . J. Eur. Math. Soc. 22 (2020), no. 3, pp. 833–875

DOI 10.4171/JEMS/934