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 \ (dim₇ K = s\), it is not hard to see that \ (dim₇ (K + K) 2s\). The main corollary of the paper states that if \ (0 0\). This information is deduced from an \ (L^6\) bound for the Fourier transforms of Frostman measures on \ (P\). If \ (0 0\), then there exists \ (= (s) > 0\) such that \ (\|\|₋^₆ (B (R) ) ^6 R^2 - (2s +) \) for all sufficiently large \ (R 1\). The proof is based on a reduction to a \ (\) -discretised point-circle incidence problem, and eventually to the \ ( (s, 2s) \) -Furstenberg set problem.