A classical theorem of Kronecker says that the sequence $(\{\alpha_1 n\}, \{\alpha_2 n\},\dots,\{\alpha_d n\})$ ($n \in \mathbb{N}$) is uniformly distributed in $[0,1)^d$ provided that $1,\alpha_1,\alpha_2,\dots,\alpha_d$ are linearly independent over $\mathbb{Q}$. More generally, for a polynomial $p(x)$ (or a $d$-tuple of polynomials $p_1(x),p_2(x),\dots,p_d(x)$) a theorem of Weyl completely describes the distribution of $\{p(n)\}$ ($n \in \mathbb{N}$) (respectively, $(\{p_1(n)\}, \{p_2(n) n\},\dots,\{p_d(n)\})$). More generally still, the distribution of bounded generalised polynomials - i.e., expressions built up using polynomials, and the operations of addition, multiplication and fractional part - is completely understood thanks to work of Bergelson and Leibman. Generalised polynomials include, for instance, expressions such as $\{\alpha n \{\beta n \} \}$.

I'm interested in the behaviour of expressions that resemble generalised polynomials, but also include division. (In the application that I have in mind it is sufficient to consider relatively simple expressions of this type - such as $\{ n g(n)/h(n) \}$ where $g,h$ are generalised polynomials bounded by $1$ and $h(n) \neq 0$ for all $n \in \mathbb{N}$.) It seems to me that nothing is known about expressions like that, although I'd be delighted to be proven wrong. With this in mind, I would like to ask about the first non-trivial instance:

Is the sequence $\{n/\{ \sqrt{2}n\} \}$ uniformly distribured in $[0,1)$? Is the set of its values at least dense in $[0,1)$? What about sequences $\{\alpha n/\{ \beta n\} \}$, where $\alpha$ is non-zero and $\beta$ is irrational?

A classical theorem of Kronecker says that the sequence $(\{\alpha_1 n\}, \{\alpha_2 n\},\dots,\{\alpha_d n\})$ ($n \in \mathbb{N}$) is uniformly distributed in $[0,1)^d$ provided that $1,\alpha_1,\alpha_2,\dots,\alpha_d$ are linearly independent over $\mathbb{Q}$. More generally, for a polynomial $p(x)$ (or a $d$-tuple of polynomials $p_1(x),p_2(x),\dots,p_d(x)$) a theorem of Weyl completely describes the distribution of $\{p(n)\}$ ($n \in \mathbb{N}$) (respectively, $(\{p_1(n)\}, \{p_2(n) n\},\dots,\{p_d(n)\})$). More generally still, the distribution of bounded generalised polynomials - i.e., expressions built up using polynomials, and the operations of addition, multiplication and fractional part - is completely understood thanks to work of Bergelson and Leibman. Generalised polynomials include, for instance, expressions such as $\{\alpha n \{\beta n \} \}$.

I'm interested in the behaviour of expressions that resemble generalised polynomials, but also include division. (In the application that I have in mind it is sufficient to consider relatively simple expressions of this type - such as $\{ n g(n)/h(n) \}$ where $g,h$ are generalised polynomials bounded by $1$ and $h(n) \neq 0$ for all $n \in \mathbb{N}$.) It seems to me that nothing is known about expressions like that, although I'd be delighted to be proven wrong. With this in mind, I would like to ask about the first non-trivial instance:

Is the sequence $\{n/\{ \sqrt{2}n\} \}$ uniformly distributed in $[0,1)$? Is the set of its values at least dense in $[0,1)$? What about sequences $\{\alpha n/\{ \beta n\} \}$, where $\alpha$ is non-zero and $\beta$ is irrational?