Arithmetic structure of the exceptional set of projections

2022/10/23 by Chen, Changhao, Miao, Zhengyan
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2210.12677

Abstract

We study the arithmetic structure of the exceptional set of projections. For any bounded subset E⊂ ℝ^d, let Ω=\ξ∈ ℝ: dim_B(E+ξE)=dim_B E\. We prove that either Ω=\0\ or Ω is a subfield of ℝ. We show that in general the statement does not hold for Hausdorff dimension and lower box dimension. Moreover, for any s∈ (0, 1] and a sequence (r_k) ⊂ ℝ, we construct a Ahlfors s-regular set E⊂ ℝ² such that for any r_k, k∈ ℕ, we have \[ dim_B \x+r_k y: (x, y)∈ E\

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