Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Regionally proximal relation of order d is an equivalence one for
  minimal systems and a combinatorial consequence   By proving the minimality of face transformations acting on the diagonal
points and searching the points allowed in the minimal sets, it is shown that
the regionally proximal relation of order $d$, $\RP^{[d]}$, is an equivalence
relation for minimal systems. Moreover, the lifting of $\RP^{[d]}$ between two
minimal systems is obtained, which implies that the factor induced by
$\RP^{[d]}$ is the maximal $d$-step nilfactor. The above results extend the
same conclusions proved by Host, Kra and Maass for minimal distal systems.
  A combinatorial consequence is that if $S$ is a dynamically syndetic subset
of $\Z$, then for each $d\ge 1$, $$\{(n_1,\...,n_d)\in \Z^d: n_1\ep_1+...
+n_d\ep_d\in S, \ep_i\in \{0,1\}, 1\le i\le d\}$$ is syndetic. In some sense
this is the topological correspondence of the result obtained by Host and Kra
for positive upper Banach density subsets using ergodic methods.

19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
18
The categories of this paper are: econ
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
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The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th