Sublacunary sets and interpolation sets for nilsequences

Let $G$ be a $k$-step nilpotent Lie group and $\Gamma\subset G$ be a discrete, cocompact subgroup. The homogeneous space $X=G/\Gamma$ is called a $k$-step nilmanifold. For any $g\in G$, the map $g:X\to X$ defined by $x\mapsto gx$ is a diffeomorphism and the pair $(X,g)$ is called a $k$-step nilsystem.

For $x\in X$ and a continuous function $F$ on $X$, the sequence $(F(g^{n}x))_{n\in\mathbb{N}}$ is called a basic $k$-step nilsequence. Examples of basic $k$-step nilsequences include $(e^{2\pi in^{k}\alpha})_{n\in\mathbb{N}}$ where $\alpha\in\mathbb{R}$, or more generally, $(e^{2\pi iP(n)})_{n\in\mathbb{N}}$ where $P\in\mathbb{R}[x]$ of degree not greater than $k$.

A sequence $\psi:\mathbb{N}\to\mathbb{C}$ is a $k$-step nilsequence if for every $\epsilon>0$, there exists a basic $k$-step nilsequence $\psi_{\epsilon}:\mathbb{N}\to\mathbb{C}$ such that $|\psi(n)-\psi_{\epsilon}(n)|<\epsilon$ for all $n\in\mathbb{N}$. In other words, a $k$-step nilsequence is a uniform limit of basic $k$-step nilsequences.

The space of $k$-step nilsequences is closed under pointwise addition, multiplication, complex conjugation and uniform limits.

More details on nilsequences can be found in [2, Section 4.3.1] and [14, Section 11.3.2].

We can always embed a $k$-step nilmanifold $X=G/\Gamma$ into a $k$-step nilmanifold $X^{\prime}=G^{\prime}/\Gamma^{\prime}$ where $G^{\prime}$ is connected and simply connected and every element of $G$ is represented in $G^{\prime}$ (for example, see [14, Corollary 26, Section 10.5.1]). Therefore, every $k$-step nilsequence can be realized from a $k$-step nilmanifold with a connected and simply connected Lie group.

For $A\subset\mathbb{N}$ and $\alpha\in\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$, let $\overline{A\alpha}$ denote the closure of $A\alpha:=\{a\alpha:a\in A\}$ in $\mathbb{T}^{d}$.

Two sets $A,B\subset\mathbb{N}$ are said to be separable by a Bohr rotation if there exists a finite dimensional torus $\mathbb{T}^{d}$ and an element $\alpha\in\mathbb{T}^{d}$ such that $\overline{A\alpha}\cap\overline{B\alpha}=\varnothing$.

Strzelecki [23] proved that all lacunary sets are $I_{0}$-sets. By Hartman-Ryll-Nardzewski characterization [13], some unions of lacunary sets are $I_{0}$-sets, for example, $\{2^{n}\}_{n\in\mathbb{N}}\cup\{2^{n}+1\}_{n\in\mathbb{N}}$. In fact, it is shown in [7] that any finite union of shifts of lacunary sets is an $I_{0}$-set, for instance, $\bigcup_{i=1}^{\ell}\{2^{n}+i\}_{n\in\mathbb{N}}$. Grow [11] constructed a class of $I_{0}$-sets which are not finite unions of lacunary sets, for example, $\{3^{n^{2}}+3^{j}:n\geq 1,(n-1)^{2}\leq j\leq n^{2}\}$ (see also Méla [19]). More information on $I_{0}$-sets can be found in the book [7] by Graham and Hare.

For $A\subset\mathbb{N}$, for a $k$-step nilsystem $(X=G/\Gamma,g)$ and $x\in X$, let $g^{A}x$ denote the set $\{g^{a}x:a\in A\}$ and let $\overline{g^{A}x}$ denote the closure of $g^{A}x$ in $X$.

Analogous to Hartman-Ryll-Nardzewski’s criterion [13], we have a characterization of $k$-step interpolation sets and its proof is identical to [16, Theorem 3.1].

Let $\pi:G\to X$ be the natural quotient map $\pi(g)=g\Gamma$. Let $1_{G}$ be the identity element of $G$ and $1_{X}=\pi(1_{G})$. In Definition 2.3, by changing the base point to $1_{X}$, we have an equivalent definition: $R\subset\mathbb{N}$ is a set of pointwise $k$-step nilrecurrence if and only if for any $k$-step nilsystem $(X,g)$, we have $1_{X}\in\overline{g^{R}1_{X}}$.

For any $k\in\mathbb{N}$, the set $k\mathbb{N}$ is a set of pointwise $k$-step nilrecurrence. Other than that, little is known about sets of pointwise nilrecurrence. However, there is a related and better-studied notion called sets of topological $k$-step nilrecurrence. A set $R\subset\mathbb{N}$ is called a set of topological $k$-step nilrecurrence if for any $k$-step nilsystem $(X,g)$ and for any $U\subset X$ open, there exists $r\in R$ such that $U\cap g^{-r}U\neq\varnothing$.³³3Usually when defining sets of topological recurrence, we restrict to minimal systems. However, as every orbit closure in nilsystems is minimal, we do not need this restriction here. It is proved in [15] that a set $R$ is a set of topological $k$-step nilrecurrence if and only if it is a set of Bohr recurrence.

It is obvious that every set of pointwise $k$-step nilrecurrence is a set of topological $k$-step nilrecurrence. But the converse is probably false (See [20].) For example, while $\{n^{2}\}_{n\in\mathbb{N}}$ is a set of topological nilrecurrence, it is not clear whether it is a set of pointwise $2$-step nilrecurrence or not.

Let $X=G/\Gamma$ be a $k$-step nilmanifold with connected and simply connected $G$. If $m$ is the dimension of $G$ and $\mathcal{G}$ is the Lie algebra of $G$, then $\mathcal{G}$ admits a base $(\xi_{1},\ldots,\xi_{m})$ satisfying the following properties:

1. (1)

   The map $\psi:\mathbb{R}^{m}\to G$ defined by

   $$\psi(t_{1},\ldots,t_{m})=\exp(t_{1}\xi_{1})\cdots\exp(t_{m}\xi_{m})$$

   is a diffeomorphism from $\mathbb{R}^{m}$ onto $G$.

2. (2)

   $\psi(\mathbb{Z}^{m})=\Gamma$.

The base $(\xi_{1},\ldots,\xi_{m})$ is called a Mal’cev basis of $G$. If $g=\psi(t_{1},\ldots,t_{m})$, then $t_{1},\ldots,t_{m}$ are called the Mal’cev coordinates of $g$ in the base $(\xi_{1},\ldots,\xi_{m})$.

For $\underline{s}=(s_{1},\ldots,s_{m})$ and $\underline{t}=(t_{1},\ldots,t_{m})\in\mathbb{R}^{m}$, write

There exist $m-1$ polynomials $P_{1},\ldots,P_{m-1}$ with $P_{i}\in\mathbb{Q}[s_{1},\ldots,s_{i},t_{1},\ldots,t_{i}]$ of degree at most $k-1$ such that for all $\underline{s},\underline{t}\in\mathbb{R}^{m}$,

We have $\psi([0,1)^{m})$ is a fundamental domain for $X$. Hence, we can identify $X$ with $[0,1)^{m}$ through the diffeomorphism $\widetilde{\psi}:X\to[0,1)^{m}$.

We then have a chain of maps:

and define $\varphi:\mathbb{R}^{m}\to[0,1)^{m}$ to be $\varphi=\tilde{\psi}\circ\pi\circ\psi$. Equivalently, for $\underline{x}\in\mathbb{R}^{m}$, there exists a unique $\underline{z}(x)\in\mathbb{Z}^{m}$ such that $\underline{x}*\underline{z}(x)\in[0,1)^{m}$. Then we can see that $\varphi(\underline{x})=\underline{x}*\underline{z}(x)$. Further details on Mal’cev bases can be found in [18, 3] and [14, Chapter 10].

We can see that $\lVert\cdot\rVert_{\mathbb{T}^{m}}$ defines a metric on $[0,1)^{m}$ which is compatible with the Euclidean metric. The reason we choose this metric only to make our computations later less cumbersome. We use $d$ to denote the metric on $X$ that comes from the metric $\lVert\cdot\rVert_{\mathbb{T}^{m}}$ on $[0,1)^{m}$ through the map $\widetilde{\psi}^{-1}$.

We use $\mathbb{N}$ to denote the set of positive integers $\{1,2,3,\ldots\}$ and use $\mathbb{T}$ to denote the torus $\mathbb{R}/\mathbb{Z}$. The notation $(b(n))_{n\in\mathbb{N}}$ represents a sequence of complex numbers and $\{r_{n}\}_{n\in\mathbb{N}}$ represents a subset of $\mathbb{N}$ with $r_{1}<r_{2}<\ldots$.

Let $f,g:\mathbb{R}\to\mathbb{R}$. By writing $f(x)=O_{a,b}(g(x))$, we mean there exist positive constants $C$ and $N_{0}$ that depend on $a$ and $b$ such that $|f(x)|<Cg(x)$ for every $x>N_{0}$.

For the rest of Section 4.1, we fix a $k$-step nilmanifold $X=G/\Gamma$ with $G$ being connected and simply connected. Assume $\dim G=m$ and the multiplication rule on the Mal’cev coordinates is the following:

where for $1\leq i\leq m-1$, the polynomial $P_{i}\in\mathbb{Q}[s_{1},\ldots,s_{i},t_{1},\ldots,t_{i}]$ has degree at most $k-1$. We also fix the maps $\psi,\pi,\widetilde{\psi}$ and $\varphi$ accordingly as they are defined in Section 2.6.

All constants and big-O terms found in this section implicitly depend on $G,\Gamma$, the choice of Mal’cev’s basis, maps $\psi,\pi,\widetilde{\psi},\varphi$, and polynomials $P_{1},\ldots,P_{m-1}$.

In the rest of Section 4.1, we fix polynomials $Q_{i,n}$ (for $i\in\{1,\ldots,m\}$ and $n\in\mathbb{N}$) found in Lemma 4.2.