18 1. Introduction
We will not discuss the proof of Theorem 1.2.7 from [GrRu2007] here,
save to say that it relies heavily on Fourier-analytic methods, and as such,
does not seem to easily extend to a general nonabelian setting. To state the
nonabelian analogue of Theorem 1.2.7, one needs multiplicative analogues of
the concept of a generalised arithmetic progression. An ordinary (symmet-
ric) arithmetic progression {−Nv,...,Nv} has an obvious multiplicative
analogue, namely a (symmetric) geometric progression
, . . . ,
} for
some generator a G. In a similar vein, if one has r commuting generators
a1,...,ar and some dimensions N1,...,Nr 0, one can form a symmetric
generalised geometric progression
(1.4) P
. . . ar
: |ni| Ni∀1 i r},
which will still be a
group. However, if the a1,...,ar do not
commute, then the set P defined in (1.4) is not quite the right concept to
use here; for instance, there it is no reason for P to be symmetric. However,
it can be modified as follows:
Definition 1.2.8 (noncommutative progression). Let a1,...,ar be elements
of a (not necessarily abelian) group G = (G, ·), and let N1,...,Nr 0.
We define the noncommutative progression P = P (a1,...,ar; N1,...,Nr)
of rank r with generators a1,...,ar and dimensions N1,...,Nr to be the
collection of all words w composed using the alphabet a1,a1
−1,...,ar,ar −1,
such that for each 1 i r, the total number of occurrences of ai and ai
combined in w is at most Ni.
Example 1.2.9. P (a, b;1, 2) consists of the elements
where each occurrence of ± can independently be set to + or −; thus
P (a, b;1, 2) can have as many as 31 elements.
Example 1.2.10. If the a1,...,ar commute, then the noncommutative pro-
gression P (a1,...,ar; N1,...,Nr) simplifies to (1.4).
Exercise 1.2.2. Let G be the discrete Heisenberg group
(1.5) G =

1 Z Z
0 0 1

and let
e1 :=

1 1 0
0 0 1

, e2 :=

1 0 0
0 0 1

be the two generators of G. Let N 1 be a sufficiently large natural number.
Show that the noncommutative progression P (e1,e2; N, N) contains all the
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