Upper bound of the lower density for a curve-free 1 set This post is asking for a reference. So in the 'Fractal Geometry' book by K.Falconer,  in the chapter 'Local structure of Fractals', there is a statement which says that,
If $F$ is a curve free 1-set in $\mathbb{R}^2$ then for almost all $x \in F$ the lower density of $F$ at $x$ is atmost $\frac{3}{4}$.
But the proof is omitted. So if anyone can refer a elaborate proof, then I will be very grateful.
 A: You have the wrong book by Falconer :D The one you are looking for is 'Geometry of fractal sets' [1], where you can find a proof in Section 3.3, 'Density and the characterization of regular $1$-sets'.
Just to add a bit of history (and more places where you can find a proof), the original proof in $\mathbb{R}^2$ is by Besicovitch [2, Theorem 14].
Later Moore extended the result (with a constant bigger than $\tfrac34$) for sets in $\mathbb{R}^n$ [3].
Preiss and Tišer then slightly improved the bound from $\tfrac34$ to $\frac{2+\sqrt{46}}{12}\approx 0.7318...$, and their proof holds in metric spaces [4].
You can find a few more references in Falconer's book.

[1] K. Falconer, The geometry of fractal sets.
Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986, xiv+162.
[2] A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points II. Math. Ann. 115 (1938), 296-329.
[3] E.F. Moore, Density ratios and $(\phi,1)$ rectifiability in $n$-space.
Trans. Amer. Math. Soc., 69 (1950), 324–334.
[4] D. Preiss, J. Tišer, On Besicovitch's $\tfrac12$-problem. J. London Math. Soc. (2) 45 (1992), no. 2, 279–287.
