# Additive properties of fractal sets on the parabola

## Authors

• Tuomas Orponen University of Jyväskylä, Department of Mathematics and Statistics

## Keywords:

Fourier transforms, additive energies, Furstenberg sets, Frostman measures

### Abstract

Let $$0 \leq s \leq 1$$, and let $$\mathbb{P} := \{(t,t^{2}) \in \mathbb{R}^{2} \colon t \in [-1,1]\}$$. If $$K \subset \mathbb{P}$$ is a closed set with $$\operatorname{dim}_{\mathrm{H}} K = s$$, it is not hard to see that $$\operatorname{dim}_{\mathrm{H}} (K + K) \geq 2s$$. The main corollary of the paper states that if $$0 < s < 1$$, then adding $$K$$ once more makes the sum slightly larger:
$$\operatorname{dim}_{\mathrm{H}} (K + K + K) \geq 2s + \epsilon,$$

where $$\epsilon = \epsilon(s) > 0$$. This information is deduced from an $$L^{6}$$ bound for the Fourier transforms of Frostman measures on $$\mathbb{P}$$. If $$0 < s < 1$$, and $$\mu$$ is a Borel measure on $$\mathbb{P}$$ satisfying $$\mu(B(x,r)) \leq r^{s}$$ for all $$x \in \mathbb{P}$$ and $$r > 0$$, then there exists $$\epsilon = \epsilon(s) > 0$$ such that
$$\|\hat{\mu}\|_{L^{6}(B(R))}^{6} \leq R^{2 - (2s + \epsilon)}$$
for all sufficiently large $$R \geq 1$$. The proof is based on a reduction to a $$\delta$$-discretised point-circle incidence problem, and eventually to the $$(s,2s)$$-Furstenberg set problem.

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Section
Articles

2023-01-02

## How to Cite

Orponen, T. (2023). Additive properties of fractal sets on the parabola. Annales Fennici Mathematici, 48(1), 113–139. https://doi.org/10.54330/afm.125826