Projections of planar sets in well-separated directions

Abstract

This paper contains two new projection theorems in the plane.First, let K⊂B(0,1)⊂R2 be a set with H∞1(K)∼1, and write πe(K) for the orthogonal projection of K into the line spanned by e∈S1. For 1/2≤s<1, writeEs:={e:N(πe(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(Es,δ)⪅δ−s. Using the polynomial method, I prove thatN(Es,r)⪅min⁡{δ−s(δr)1/2,r−1},δ≤r≤1. I construct examples showing that the exponents in the bound are sharp for δ≤r≤δs.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 Sp:={e2πik/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, then1p∑e∈SpN(πe(K),δ)≥δ−s for all small enough δ>0. In particular, dim‾Bπe(K)≥s for some e∈Sp.