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.



Examples of probability measures with `fake' decay
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 $l$.


