Strongly central sets and sets of polynomial returns mod 1

Central sets in $\mathbb{N}$ were introduced by Furstenberg and are known to have substantial combinatorial structure. For example, any central set contains arbitrarily long arithmetic progressions, all finite sums of distinct terms of an infinite sequence, and solutions to all partition regular systems of homogeneous linear equations. We introduce here the notions of strongly central and very strongly central, which as the names suggest are strictly stronger than the notion of central. They are also strictly stronger than syndetic, which in the case of $\mathbb{N}$ means that gaps are bounded.

Given $x\in \mathbb{R}$, let $w(x)=x-\lfloor x+\frac {1}{2}\rfloor$. Kronecker's Theorem says that if $1,\alpha _1,\alpha _2,\ldots ,\alpha _v$ are linearly independent over $\mathbb{Q}$ and $U$ is a nonempty open subset of $(-\frac {1}{2},\frac {1}{2})^v$, then $\{x\in \mathbb{N}:(w(\alpha _1 x),\ldots ,w(\alpha _v x))\in U\}$ is nonempty, and Weyl showed that this set has positive density. We show here that if $\overline 0$ is in the closure of $U$, then this set is strongly central. More generally, let $P_1,P_2,\ldots ,P_v$ be real polynomials with zero constant term. We show that

$$\{x\in \mathbb{N}:(w(P_1(x)),\ldots ,w(P_v(x)))\in U\}$$

is nonempty for every open $U$ with $\overline 0\in c\ell U$ if and only if it is very strongly central for every such $U$ and we show that these conclusions hold if and only if any nontrivial rational linear combination of $P_1,P_2,\ldots ,P_v$ has at least one irrational coefficient.