1.3. Linear patterns 55
the affine-linear span (over Q) of the forms in any of these classes. The
Cauchy-Schwarz complexity of a system is defined to be the least such s
with this property, or if no such s exists.
The adjective “Cauchy-Schwarz” (introduced by Gowers and Wolf
[GoWo2010]) may be puzzling at present, but will be motivated later.
This is a somewhat strange definition to come to grips with at first, so
we illustrate it with some examples. The system of forms x, y, x + y is of
complexity 1; given any form here, such as y, one can partition the remaining
forms into two classes, namely {x} and {x + y}, such that y is not in the
affine-linear span of either. On the other hand, as y is in the affine linear
span of {x, x + y}, the Cauchy-Schwarz complexity is not zero.
Exercise 1.3.13. Show that for any k 2, the system of forms x, x +
y, . . ., x + (k 1)y has complexity k 2.
Exercise 1.3.14. Show that a system of non-constant forms has finite
Cauchy-Schwarz complexity if and only if no form is an affine-linear combi-
nation of another.
There is an equivalent way to formulate the notion of Cauchy-Schwarz
complexity, in the spirit of the change of variables mentioned earlier. Define
the characteristic of a finite abelian group G to be the least order of a
non-identity element.
Proposition 1.3.3 (Equivalent formulation of Cauchy-Schwarz complex-
ity). Let ψ1,...,ψt :
G be a system of affine-linear forms. Suppose
that the characteristic of G is sufficiently large depending on the coefficients
of ψ1,...,ψt. Then ψ1,...,ψt has Cauchy-Schwarz complexity at most s if
and only if, for each 1 i t, one can find a linear change of variables x =
Li(y1,...,ys+1,z1,...,zm) over Q such that the form
ψ i(Li(y1,...,ys+1,
z1,...,zm)) has non-zero y1,...,ys+1 coefficients, but all the other forms
ψ with j = i have at least one vanishing
y1,...,ys+1 coefficient, and
ψ :
Q is the linear form induced by the
integer coefficients of ψi.
Proof. To show the “only if” part, observe that if 1 i t and Li is as
above, then we can partition the ψj, j = i into s + 1 classes depending on
which yk coefficient vanishes for k = 1,...,s + 1 (breaking ties arbitrarily),
and then ψi is not representable as an affine-linear combination of the forms
from any of these classes (here we use the large characteristic hypothesis).
Conversely, suppose ψ1,...,ψt has Cauchy-Schwarz complexity at most s,
and let 1 i s. We can then partition the j = i into s + 1 classes
A1,..., As+1, such that ψi cannot be expressed as an affine-linear combina-
tion of the ψj from Ak for any 1 k s + 1. By duality, one can then find
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