Numbers $a$ that are the sum of the fractional parts $\{x^2\} + \{x\}$ for some $x$

Question

Are there infinitely many rational numbers $a\in\mathbb{Q}$ with the following property:

$\{x^2\}+\{x\}=a$ for infinitely many $x\in\mathbb{Q}^+$

Answer 1

Unless I'm missing something, for every number $a=1/n+1/n^2$ there are infinitely many $x$ with claimed property: let's take $x=kn+1/n$ for any $k\in\Bbb N$. Then $\{x\}=1/n$, $x^2=k^2n^2+2k+1/n^2, \{x^2\}=1/n^2$, $\{x^2\}+\{x\}=1/n^2+1/n=a$.

Answer 2

Hint: what is the maximum the left side can be? If $x$ is a rational solution, can you find another one?