# Examples of probability measures with `fake' decay

To be concise, I am wondering whether there are natural examples of probability measures $\mu$ compactly supported on the real line which satisfy $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ with $|I| = l_n$, for some $0 \leq \alpha \leq 1$ and a certain sequence $l_1 \geq l_2 \geq \dots \to 0$, but for which the general inequality $\mu(I) \lesssim |I|^\alpha$ fails.

Often, in order to lower bound the Hausdorff dimension of a certain set $X$ by a constant $\alpha$, one applies Frostman's lemma, finding a natural probability measure $\mu$ supported on $X$ with $\mu(I) \lesssim |I|^\alpha$. Often, one constructs $X$ as a fractal, as a limit of sets $X_n$, each a union of intervals of length $l_n$ (i.e. $X_1 \supset X_2 \supset \dots \to X$). Often, one obtains $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ of length $l_n$, and then tries to 'cheat out' the general bound from these estimates. My question is whether there are natural counterexamples of 'fake' $\alpha$ dimensional probability measures for which one can obtain the bound $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ of length $l_n$, where $l_1 \geq l_2 \geq \dots$ is a sequence converging to zero, but for which the bound $\mu(I) \lesssim |I|^\alpha$ does not hold for all intervals.

• This reminds me of the faithfulness notion for Hausdorff dimension. You might want to look here, especially Theorem 1. – zhoraster Jul 16 '18 at 11:56

Later on in my research I answered this question in the negative:

Consider a sequence of dyadic scales $$\{ r_k \}$$, such that $$r_k/r_{k+1} \geq 4$$. Consider a sequence of dyadic lengths $$\{ r_k \}$$, and construct a set $$X$$ by a Cantor-type decomposition, defined as $$\lim_{k \to \infty} X_k$$ where $$X_k$$ is a union of side-length $$r_k$$ intervals. We set $$X_0 = [0,1]$$, and $$r_0 = 1$$. Given $$X_k$$, which is a union of sidelength $$r_k$$ intervals, we define $$X_{k+1}$$ arbitrarily, such that for any sidelength $$r_k$$ interval in $$X_k$$, $$X_{k+1}$$ contains a single sidelength $$r_{k+1}^{1/2}$$ interval. If we let $$N_k$$ to be the covering number of $$X_k$$ by $$r_k$$ intervals, then $$N_0 = 1$$, and $$N_{k+1} = N_k(r_k/r_{k+1}^{1/2})$$. Thus

$$N_k = \frac{\left( r_1 \dots r_{k-1} \right)^{1/2}}{r_k^{1/2}}$$

If we set $$\mu(I) = 1/N_k$$ for each side-length $$r_k$$ interval $$I$$ selected in $$X_k$$, and $$\mu(I) = 0$$ otherwise, then $$\mu$$ satisfies the mass distribution principle and thus extends to a Borel probability measure. If $$l_k = r_k^{1/2}$$, then for any side-length $$l_{k+1}^{1/2}$$ interval, either $$\mu(I) = 0$$ or

$$\mu(I) = \frac{l_{k+1}}{(r_1 \dots r_k)^{1/2}} \leq \frac{l_{k+1}}{r_k^{k/2}}.$$

If $$r_{k+1} \leq r_k^{4k}$$, then $$l_{k+1}^{1/4} = r_{k+1}^{1/8} \leq r_k^{k/2}$$, so $$\mu(I) \leq l_k^{3/4}$$ for each index $$k$$ and side-length $$l_k$$ interval $$I$$. Thus at the scales $$l_k$$, the set $$X$$ looks like it has dimension at least $$3/4$$. But as $$k \to \infty$$,

$$H^{1/2}_{r_k}(X) \leq N_k \cdot r_k^{1/2} \leq (r_1 \dots r_{k-1})^{1/2} \to 0$$

Thus $$\dim(X) \leq 1/2$$, so we cannot possibly obtain a general bound $$\mu(I) \leq_\varepsilon l^{3/4-\varepsilon}$$ for all scales $$l$$.