1.2. Roth’s theorem 41

Now if r is one of the periods in the above lemma, we have

En∈[N]|(fstr + fsml)(n + r) − (fstr + fsml)(n)| ε

and thus by shifting

En∈[N]|(fstr + fsml)(n + 2r) − (fstr + fsml)(n + r)| ε

and so by the triangle inequality

En∈[N]|(fstr + fsml)(n + 2r) − (fstr + fsml)(n)| ε.

Putting all this together using the triangle and H¨ older inequalities, we obtain

En∈[N](fstr + fsml)(n)(fstr + fsml)(n + r)(fstr + fsml)(n + 2r) ≥

δ3

− O(ε).

Thus, if ε is suﬃciently small depending on δ, we have

En∈[N](fstr + fsml)(n)(fstr + fsml)(n + r)(fstr + fsml)(n + 2r)

δ3

for

J,ε

N values of r, and thus

Λ(fstr + fsml,fstr + fsml,fstr + fsml)

δ,M

1;

if we then set F to be a suﬃciently rapidly growing function (depending

on δ), we obtain the claim from (1.15). This concludes the proof of Roth’s

theorem.

Exercise 1.2.12. Use the energy increment method to establish a different

proof of Exercise 1.2.7. (Hint: For the multiple recurrence step, use a

pigeonhole principle argument rather than an appeal to equidistribution

theory.)

We now briefly indicate how to translate the above arguments into the

ultralimit setting. We first need to construct an important measure on limit

sets, namely Loeb measure.

Exercise 1.2.13 (Construction of Loeb measure). Let N be an unbounded

natural number. Define the Loeb measure μ(A) of a limit subset A of [N]

to be the quantity st(|A|/N), thus for instance a set of cardinality o(N) will

have Loeb measure zero.

(i) Show that if a limit subset A of [N] is partitioned into countably

many disjoint limit subsets An, that all but finitely many of the An

are empty, and so μ(A) = μ(A1) + · · · + μ(An).

(ii) Define the outer measure μ∗(A) of a subset A of [N] (not necessarily

a limit subset) to be the infimum of

∑

n

μ(An), where A1,A2,...

is a countable family of limit subsets of [N] that cover A, and call

a subset of [N] null if it has zero outer measure. Call a subset

Loeb measurable if it differs from a limit set by a null set. Show

that there is a unique extension of Loeb measure μ from limit sets

to Loeb measurable sets that is a countably additive probability