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## Hausdorff dimension of certain sets

## 1. Generalized Cantor set:

Fix . Choose (small) and (large) such that . Fix .

Let be the union of intervals of length . Let be the natural measure (normalized one dimensional Lebesgue measure restricted to ) arises from .

Let be the union of intervals of length and be its natural measure.

Similarly for each let be the union of intervals of length and be its natural measure.

We consider the generalized Cantor set .

(Exercise: Then is compact perfect and nowhere dense set.)

Also note that,

- weakly converges to a measure supported in .
- is locally compact and , the space of compactly supported continuous function on is a separable topological vector space.
- By the application of Riesz representation theorem, space of continuous function vanishing at infinity is same as the space of Radon measures. (All the above ‘s are Radon measure, that is locally finite and inner regular measure)
- By the application of Banach-Alaoglu theorem, is a weakly convergent sequence.

- (enough to prove ).
- By the definition of , for there exists large such that . Then

and hence .

- Since is compact, if is a Borel cover of then we can assume that is a finite Borel cover. Also since is totally disconnected, can be assumed to be finite Borel cover, with ‘s disjoint open intervals and their end points are in the complement of .Let be such a cover. Then for each let be the smallest integer such that has at least one interval of length . Note that there cannot be more than consecutive intervals of length in (otherwise, it contradicts the choice of to be the smallest). Let be the number of consecutive intervals of length in . Then . Hence diameter of .
We know that each interval of length has intervals of length and intervals of length . Similarly each interval of length has intervals of length for . Thus intervals of length has intervals of length . There are totally intervals of length and since ‘s are disjoint and finite, we have for large (for all ).

Also we have . Hence

Thus .

- By the definition of , for there exists large such that . Then

(to be updated) This blog is loosely based on the notes prepared while giving a lecture series during January 2017-April 2017 at IISER - Bhopal. The definitions, the notations, proofs, theorems are from the references: 1) W. F. Donoghue; Distributions and Fourier Transforms, Acad. Press, 1969. MR3363413 2) Pertti Mattila, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. MR1333890 3) P. Mattila. Fourier Analysis and Hausdorff Dimension, Cambridge University Press, Cambridge, 2015. 4) Wolff T.H.: Lectures on Harmonic Analysis. University Lecture Series, 29. Amer. Math. Soc., Providence, RI (2003)