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Let $x$ be a large positive real number. Let $I$ be an interval -- say, $I=[1,\sqrt{\epsilon x}]$. Let $n$ range over the integers in $I$, or over the intersection of $I$ and an arithmetic progression of small modulus.

Can one say anything sensible about the distribution of the fractional part $\{x/n^2\}$? Equivalently: can one give non-trivial bounds on $\sum_{n\in I\cap \mathbb{Z}} e(x/n^2)$, where $e(r) = e^{2\pi i r}$?

(Motivation: bounds on the error term of the asymptotic formula for the number of square-free numbers up to $x$.)

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I'm not sure but I'll point out two things which are hopefully somehow useful (and if they're not I'll just delete it later).

First, for the range close to $\sqrt x$: you have \[ \int _{\epsilon \sqrt x}^{\sqrt x}e(x/t^2)dt=\sqrt x\int _{1}^{1/\epsilon ^2}\frac {e(u)du}{u^{3/2}}\] and there's no reason why the corresponding sum shouldn't approximate to its integral (right, I think?). So there you get the actual size of the sum.

Second, for the range below $\sqrt x$: for $t\approx x^{1/2-\delta }$ you have $d/dt(x/t^2)\approx 1/x^{1/2-3\delta }$ so you have (again "I think") \[ \sum _{n\sim x^{1/2-\delta }}e(x/n^2)\ll x^{1/2-3\delta }\] by e.g. Corollary 8.11 of Iwaniec-Kowalski (again after having approximated to to the sum with an integral obviously).

So, dunno. Maybe it depends exactly what growth $\epsilon $ has?

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You could use a sieve to split into bilinear forms then apply a generalization of Theorem 14 of

https://arxiv.org/pdf/1211.4184.pdf

to estimate the resulting sums

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