Let $\mu$ be a continuous measure on $[0,1]$ (i.e. each individual point has $0$ measure). As usual, denote by $\hat\mu(n)=\int\_0^1e^{2\pi inx}d\mu(x)$ the Fourier transform of $\mu$, and let $\lfloor x\rfloor$ denote the [floor](https://en.wikipedia.org/wiki/Floor_and_ceiling_functions) of $x\in\mathbb R$. Is it true that
$$\lim\_{N\to\infty}sup\_{M\in\mathbb N}\frac1N\sum\_{n=M}^{M+N}\left|\hat\mu\Big(\big\lfloor n^{3/2}\big\rfloor\Big)\right|=0~?$$


Observe that if $\mu$ is absolutely continuous, then the answer is yes by the Riemann-Lebesgue lemma.


If instead of $\hat\mu(\lfloor n^{3/2}\rfloor)$ we consider $\hat\mu(n)$, then the answer is again yes (it follows from Wiener's theorem). 


If we don't take a supremum over all shifts of the interval $\{1,\dots,N\}$, then the result is again well known because the sequences $n\mapsto\lfloor n^{3/2}\rfloor x$ are uniformly distributed for any irrational $x$.


The motivation for this question comes from the fact that 
$$\lim\_{N\to\infty}sup\_{M\in\mathbb N}\frac1N\sum\_{n=M}^{M+N}\left|\hat\mu\big(n^2\big)\right|=0.$$
This follows easily from the fact that the sequences $n\mapsto n^2 x$ are *well distributed* for every irrational $x$, but this is not the case for the sequences $n\mapsto\lfloor n^{3/2}\rfloor x$.



Average decay of Fourier coefficients of continuous measures along the sequence $\lfloor n^{3/2}\big\rfloor$
Really, you have almost answered it yourself, just left the very final words out.


Take $M=4N^4$. Then $\lfloor (M+n)^{3/2}\rfloor=8N^6+3N^2n$ for $n=1,\dots,N$, so $\frac 1N\sum\_{n=1}^N e^{-2\pi i \lfloor (M+n)^{3/2}\rfloor x}$ is $1$ when $x=q/N^2$ and nearly $1$ on a small open neighborhood $U\_N$ of those points. Now take some very fast increasing sequence of $N$ so that $\cap\_N U\_N$ contains a Cantor-type set and put the measure on it. The exact formulae do not matter, of course. What mattered a bit was having long linear pieces of arbitrarily large slope, but even that wasn't crucial. As you noticed yourself, the (particular kind of) absence of good distribution was the key.


