REVIEW 3 cited by
On the projections of Ahlfors regular sets in the plane
Not yet reviewed by Pith; the record is open.
This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.
SPECIMEN: schema-true, not a live event
T0 review · schema-true
One-sentence machine reading of the paper's core claim.
pith:XXXXXXXX · record.json · timestamp
abstract
This paper contains the following $\delta$-discretised projection theorem for Ahlfors regular sets in the plane. For all $C,\epsilon > 0$ and $s \in [0,1]$, there exists $\kappa > 0$ such that the following holds for all $\delta > 0$ small enough. Let $\nu$ be a Borel probability measure on $S^{1}$ satisfying $\nu(B(x,r)) \leq Cr^{\epsilon}$ for all $x \in S^{1}$ and $r > 0$. Let $K \subset B(1) \subset \mathbb{R}^{2}$ be Ahlfors $s$-regular with constant at most $C$. Then, there exists a vector $\theta \in \mathrm{spt\,} \nu$ such that $$|\pi_{\theta}(F)|_{\delta} \geq \delta^{\epsilon - s}$$ for all $F \subset K$ with $|F|_{\delta} \geq \delta^{\kappa - s}$. Here $\pi_{\theta}(z) = \theta \cdot z$ for $z \in \mathbb{R}^{2}$.
Forward citations
Cited by 3 Pith papers
-
Projections of self-affine sets onto lines
Under strong pinching and strong irreducibility of the linear parts, every line projection of a self-affine set attains the expected dimension; in the plane, strong irreducibility alone suffices.
-
Furstenberg set theorem for transversal families of functions
A sharp dimension bound for Furstenberg sets built from transversal families of graphs, with an application to Fourier decay of fractal measures on convex curves.
-
Sum-product phenomena for Ahlfors-regular sets
Ahlfors regular sets of dimension s ≤ 1/2 satisfy Nδ(A+A) + Nδ(AA) ≥ δ^{-4s/3+η}, the fractal analogue of Solymosi's 4/3 bound.
Discussion (0). Continue with ORCID to comment.