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 :

Gd

→ 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

˙j(Li(y1,...,ys+1,z1,...,zm))

ψ with j = i have at least one vanishing

y1,...,ys+1 coeﬃcient, and

˙i

ψ :

Qd

→ 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