I suppose this is *une cause perdue*, but it would be nice if the following held.

Let $\theta$ be an irrational, and let $c_m = {{2m}\choose m}$, the central binomial. For a real number $r$, let $d(r)$ denote the distance from $r$ to the nearest integer (often indicated by $\| r \|$). Is it true that for every irrational $\theta$, the Cesaro sums behave reasonably well: $$ (*) \quad \limsup_{N\to \infty} \frac 1N \sum_{m=1}^{N} d(c_m \theta) > 0 \,? $$ Equivalently, does there exist $c >0$---depending on $\theta$---such that $$ \limsup_{N\to \infty} \frac{\left| \left\{m \leq N\left.\right| d(c_m \theta) > c\right\}\right|}N > 0\,? $$

For almost all $\theta$, $d(c_m\theta)$ is uniformly distributed, but not for all irrational $\theta$ (since the ratio of consecutive coefficients exceeds $1$). I don't know much more about it. It's still possible (although seemingly remote) that (*) holds for all irrational $\theta$. We can also replace the central binomial terms by the Catalan numbers and the questions are probably equivalent. (An affirmative answer to either would be fine.)

A weaker version would be whether (for all irrational $\theta$) $$ \limsup_{N\to \infty} \frac{\left| \left\{m \leq N\left.\right| d(c_m \theta) > c\right\}\right|}{N^{1/2 + \delta}} = \infty $$ for some $\delta > 0$ (depending on $\theta$); an affirmative answer would be useful.