Abstract
We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If K is a number field with ring of integers
$\mathcal{O}_K$
and
$E \subseteq \mathcal{O}_K$
has positive upper Banach density
$d^*(E) = \delta > 0$
, we show, inter alia:
(1)
if
$p(x) \in K[x]$
is an intersective polynomial (i.e., p has a root modulo m for every
$m \in \mathcal{O}_K$
) with
$p(\mathcal{O}_K) \subseteq \mathcal{O}_K$
and
$r, s \in \mathcal{O}_K$
are distinct and nonzero, then for any
$\varepsilon > 0$
, there is a syndetic set
$S \subseteq \mathcal{O}_K$
such that for any
$n \in S$
,
\begin{align*} d^* \left( \left\{ x \in \mathcal{O}_K \;:\; \{x, x + rp(n), x + sp(n)\} \subseteq E \right\} \right) > \delta^3 - \varepsilon. \end{align*}
Moreover, if
${s}/{r} \in \mathbb{Q}$
, then there are syndetically many
$n \in \mathcal{O}_K$
such that
\begin{align*} d^* \left( \left\{ x \in \mathcal{O}_K \;:\; \{x, x + rp(n), x + sp(n), x + (r+s)p(n)\} \subseteq E \right\} \right) > \delta^4 - \varepsilon; \end{align*}
(2)
if
$\{p_1, \dots, p_k\} \subseteq K[x]$
is a jointly intersective family (i.e.,
$p_1, \dots, p_k$
have a common root modulo m for every
$m \in \mathcal{O}_K$
) of linearly independent polynomials with
$p_i(\mathcal{O}_K) \subseteq \mathcal{O}_K$
, then there are syndetically many
$n \in \mathcal{O}_K$
such that
\begin{align*} d^* \left( \left\{ x \in \mathcal{O}_K \;:\; \{x, x + p_1(n), \dots, x + p_k(n)\} \subseteq E \right\} \right) > \delta^{k+1} - \varepsilon. \end{align*}
These two results generalise and extend previous work of Frantzikinakis and Kra [21] and Franztikinakis [19] on polynomial configurations in
$\mathbb{Z}$
and build upon recent work of the authors and Best [2] on linear patterns in general abelian groups. The above combinatorial results follow from multiple recurrence results in ergodic theory via a version of Furstenberg’s correspondence principle. The ergodic-theoretic recurrence theorems require a sharpening of existing tools for handling polynomial multiple ergodic averages. A key advancement made in this paper is a new result on the equidistribution of polynomial orbits in nilmanifolds, which can be seen as a far-reaching generalisation of Weyl’s equidistribution theorem for polynomials of several variables:
(3)
let
$d, k, l \in \mathbb{N}$
. Let
$(X, \mathcal{B}, \mu, T_1, \dots, T_l)$
be an ergodic, connected
$\mathbb{Z}^l$
-nilsystem. Let
$\{p_{i,j} \;:\; 1 \le i \le k, 1 \le j \le l\} \subseteq \mathbb{Q}[x_1, \dots, x_d]$
be a family of polynomials such that
$p_{i,j}\left( \mathbb{Z}^d \right) \subseteq \mathbb{Z}$
and
$\{1\} \cup \{p_{i,j}\}$
is linearly independent over
$\mathbb{Q}$
. Then the
$\mathbb{Z}^d$
-sequence
$\left( \prod_{j=1}^l{T_j^{p_{1,j}(n)}}x, \dots, \prod_{j=1}^l{T_j^{p_{k,j}(n)}}x \right)_{n \in \mathbb{Z}^d}$
is well-distributed in
$X^k$
for every x in a co-meager set of full measure.