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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.

Akash Banerjee
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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.

Del
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  • Thank you so much for such a elaborate answer. I found the 'better' Falconer book (XD) just tonight. But the other references are awesome. Very helpful. – Akash Banerjee May 30 '21 at 16:38
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    @Akash Banerjee: Regarding the 1938 paper by Besicovitch, see my answer to What is the smallest constant that has explicitly appeared in a published paper? --- To show how differently irregular $1$-sets in the plane behaved, Besicovitch proved in 1928 that the lower $1$-density of an irregular $1$-set $F$ in the plane is bounded below $1$ at almost all (Lebesgue measure) points in $F$. The bound that Besicovitch obtained in 1928 was $1 - 10^{-2576}$. Better bounds were later found . . . – Dave L. Renfro May 30 '21 at 17:12
  • @Dave L. Renfro : Thanks again. I will check your answer. I didn't knew that there is also a lower bound for the lower 1- density. Thank you. – Akash Banerjee May 30 '21 at 19:41