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Projections of planar sets in well-separated directions.
- Source :
-
Advances in Mathematics . Jul2016, Vol. 297, p1-25. 25p. - Publication Year :
- 2016
-
Abstract
- This paper contains two new projection theorems in the plane. First, let K ⊂ B ( 0 , 1 ) ⊂ R 2 be a set with H ∞ 1 ( K ) ∼ 1 , and write π e ( K ) for the orthogonal projection of K into the line spanned by e ∈ S 1 . For 1 / 2 ≤ s < 1 , write E s : = { e : N ( π e ( K ) , δ ) ≤ δ − s } , where N ( A , r ) is the r -covering number of the set A . It is well-known – and essentially due to R. Kaufman – that N ( E s , δ ) ⪅ δ − s . Using the polynomial method, I prove that N ( E s , r ) ⪅ min { δ − s ( δ r ) 1 / 2 , r − 1 } , δ ≤ r ≤ 1 . I construct examples showing that the exponents in the bound are sharp for δ ≤ r ≤ δ s . The second theorem concerns projections of 1-Ahlfors–David regular sets. Let A ≥ 1 and 1 / 2 ≤ s < 1 be given. I prove that, for p = p ( A , s ) ∈ N large enough, the finite set of unit vectors S p : = { e 2 π i k / p : 0 ≤ k < p } has the following property. If K ⊂ B ( 0 , 1 ) is non-empty and 1-Ahlfors–David regular with regularity constant at most A , then 1 p ∑ e ∈ S p N ( π e ( K ) , δ ) ≥ δ − s for all small enough δ > 0 . In particular, dim ‾ B π e ( K ) ≥ s for some e ∈ S p . [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00018708
- Volume :
- 297
- Database :
- Academic Search Index
- Journal :
- Advances in Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 115264335
- Full Text :
- https://doi.org/10.1016/j.aim.2016.04.001