Packing dimension via upper averaged Fourier dimension (conjecture)
Prove that for every Borel set X ⊆ R^d, the packing dimension dim_P(X) equals the supremum over all finite Borel measures μ supported on X of Strichartz’s upper averaged Fourier dimension at θ=1, i.e., dim_P(X) = sup{ \overline{F}_μ(1) : spt(μ) ⊆ X }, where \overline{F}_μ(1) = limsup_{R→∞} [ log( R^{-d} ∫_{|z|≤R} |\hat μ(z)|^2 dz ) / (−log R) ]. Equivalently, show that dim_P(X) equals inf{ β > 0 : for all μ supported on X, ∫_{|z|≤R} |\hat μ(z)|^2 dz ≳_{μ} R^{d−β} }.
References
We are tempted by the following conjecture, which we have been unable to prove. Recall we write $ X$ for the packing dimension of $X$. \begin{conj} Let $X \subseteq R$ be a Borel set. Then \begin{align*} X &= \sup { \overline{F}\mu(1) : \textup{spt}(\mu) \subseteq X }\ &= \inf\left{ \beta 0 : \textup{ for all $\mu$ on $X$ } \int{|z| R} |\widehat \mu (z)|{2} \, dz \gtrsim_\mu R{d-\beta} \right}. \end{align*} \end{conj} Note that in the above conjecture, the second equality is immediate from the definition of $\overline{F}_\mu(1)$ and so only the first inequality is unknown.