1.3. Linear patterns 45
that such Bohr sets were regular (this condition is closely related to the
Fourier measurability condition used in the energy increment argument).
With this modification to the original Roth argument, the bound was low-
ered to N0
δ−O(1/δ2),
or equivalently obtaining length three progressions
in sets of density much larger than log log N/ log N. Even more recently,
this argument was combined with the Szemer´ edi-Heath-Brown argument by
Bourgain [Bo2008], and refined further by Sanders [Sa2010], to obtain the
further improvement of N0
exp(O(δ−4/3−o(1))),
and then (by a somewhat
different argument of Sanders [Sa2010]) of N0
exp(O(δ−1−o(1))).
This is
tantalisingly close to the k = 3 case of an old conjecture of Erd¨ os that asserts
that any subset of the natural numbers whose sums of reciprocals diverge
should have infinitely many arithmetic progressions of length k for any k.
To establish the k = 3 case from quantitative versions of Roth’s theorem,
one would basically need a bound of the form N0
exp(δ−1+c)
for some
c 0 (or the ability to obtain progressions in sets of density 1/
log1+c
N).
Very recently, a bound of this shape has been obtained in the bounded
characteristic case; see [BaKa2011].
On the other hand, there is an old counterexample of Behrend [Be1946]
(based ultimately on the observation that a sphere in a high-dimensional lat-
tice
Zd
does not contain any arithmetic progressions of length three) which
shows that N0 must be at least
exp(log2(1/δ))
(in particular, it must
be super-polynomial in δ); equivalently, it is known that there are subsets
of [N] of density about exp(−c

log N) with no arithmetic progressions of
length three. For the sharpest results in this direction, see [El2008] and
[GrWo2008].
The question of refining the bounds is an important one, as it tends to
improve the technological understanding of current methods, as well as shed
light on their relative strengths and weaknesses. However, this comes at the
cost of making the arguments somewhat more technical, and so we shall not
focus on the sharpest quantitative results in this section.
1.3. Linear patterns
In Section 1.2, we used (linear) Fourier analysis to control the number of
three-term arithmetic progressions a, a + r, a + 2r in a given set A. The
power of the Fourier transform for this problem ultimately stemmed from
the identity
En,r∈Z/N
Z
1A(n)1A(n + r)1A(n + 2r)
=
α∈
1
N
Z/Z
ˆA(α)ˆA(−2α)ˆA(α)
1 1 1
(1.17)
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