Additive properties of fractal sets on the parabola
Let \(0 ≤ s ≤ 1\), and let \(P := {(t,t^2) R^2 t [-1,1]}\). If \(K P\) is a closed set with \(dim_H K = s\), it is not hard to see that \(dim_H (K + K) ≥ 2s\). The main corollary of the paper states that if \(0 < s < 1\), then adding \(K\) once more makes the sum slightly larger: \( dim_H (K + K + K) ≥ 2s + ε,\) where \(ε = ε(s) > 0\). This information is deduced from an \(L^6\) bound for the Fourier transforms of Frostman measures on \(P\). If \(0 < s < 1\), and \(μ\) is a Borel measure on \(P\) satisfying \(μ(B(x,r)) ≤ r^s\) for all \(x P\) and \(r > 0\), then there exists \(ε = ε(s) > 0\) such that \(\|μ\|_L^6(B(R))^6 ≤ R^2 - (2s + ε)\) for all sufficiently large \(R ≥ 1\). The proof is based on a reduction to a \(δ\)-discretised point-circle incidence problem, and eventually to the \((s,2s)\)-Furstenberg set problem. , number=1, journal=Annales Fennici Mathematici, author=Orponen, Tuomas, year=2023, month=Jan., pages=113–139