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Fix $\epsilon>0$. For a finite set of arbitrary-degree polynomials with integer constant term, $p_1(x), ..., p_m(x)\in \mathbb{R}[x]$ is it possible to find an $n\in \mathbb{N}$ such that $$\max_{i=1,...,m}||p_i(n)||<\epsilon$$ where $||\cdot||$ denotes the distance to the nearest integer?

In the book Small Fractional Parts of Polynomials this is listed as an open question for degree greater than 4, but the book is from 1977. Research has continued since then on small fractional parts of polynomials. Has progress improved on this problem, or is it still open?

Fix $\epsilon>0$. For a finite set of arbitrary-degree polynomials with integer constant term, $p_1(x), ..., p_m(x)\in \mathbb{R}[x]$ is it possible to find an $n\in \mathbb{N}$ such that $$\max_{i=1,...,m}||p_i(n)||<\epsilon$$ where $||\cdot||$ denotes the distance to the nearest integer?

In the description of the book Small Fractional Parts of Polynomials a problem of this type is listed as an open question for degree greater than 4, but the book is from 1977. Research has continued since then on small fractional parts of polynomials. Has progress improved on this problem, or is it still open?

Updates: Discussion in the comments indicates that this can be proved by Weyl's equidistribution criterion. While this proves the result for $\mathbb{Q}$-linearly independent collections $\{p_1(n),...,p_k(n), 1, n, n^2,...\}$ it does not prove it when the collection is linearly dependent, since equidistribution no longer holds in that case. How can we generalize to when the polynomials may be $\mathbb{Q}$-linearly dependent?

Fix $\epsilon>0$. For a finite set of arbitrary-degree polynomials with integer constant term, $p_1(x), ..., p_m(x)\in \mathbb{R}[x]$ is it possible to find an $n\in \mathbb{N}$ such that $$\max_{i=1,...,m}[p_i(n)]<\epsilon$$ where $[\cdot]$ denotes the nearest integer function?

In the book Small Fractional Parts of Polynomials this is listed as an open question for degree greater than 4, but the book is from 1977. Research has continued since then on small fractional parts of polynomials. Has progress improved on this problem, or is it still open?

Fix $\epsilon>0$. For a finite set of arbitrary-degree polynomials with integer constant term, $p_1(x), ..., p_m(x)\in \mathbb{R}[x]$ is it possible to find an $n\in \mathbb{N}$ such that $$\max_{i=1,...,m}||p_i(n)||<\epsilon$$ where $||\cdot||$ denotes the distance to the nearest integer?

In the book Small Fractional Parts of Polynomials this is listed as an open question for degree greater than 4, but the book is from 1977. Research has continued since then on small fractional parts of polynomials. Has progress improved on this problem, or is it still open?

Fix $\epsilon>0$. For a finite set of arbitrary-degree polynomials with integer constant term, $p_1(x), ..., p_m(x)\in \mathbb{R}[x]$ is is possible to find an $n\in \mathbb{N}$ such that $$\max_{i=1,...,m}[p_i(n)]<\epsilon$$ where $[\cdot]$ denotes the nearest integer function?

In the book Small Fractional Parts of Polynomials this is listed as an open question for degree greater than 4, but the book is from 1977. Research has continued since then on small fractional parts of polynomials. Has progress improved on this problem, or is it still open?

Fix $\epsilon>0$. For a finite set of arbitrary-degree polynomials with integer constant term, $p_1(x), ..., p_m(x)\in \mathbb{R}[x]$ is it possible to find an $n\in \mathbb{N}$ such that $$\max_{i=1,...,m}[p_i(n)]<\epsilon$$ where $[\cdot]$ denotes the nearest integer function?

In the book Small Fractional Parts of Polynomials this is listed as an open question for degree greater than 4, but the book is from 1977. Research has continued since then on small fractional parts of polynomials. Has progress improved on this problem, or is it still open?