1.2. Roth’s theorem 33
Exercise 1.2.3. Show that if N is an unbounded limit natural number, and
A ⊂ [N] is a limit subset of positive density (st |A|/N) = δ 0 with no
arithmetic progressions of length three, then there exists a limit real α such
that |En∈[N](1A(n) − δ )e(−αn)| 1.
Exercise 1.2.4. If N is an unbounded limit natural number, and α is a
limit real, show that one can partition [N] =
Pj ∪ E, where J is a
limit natural number, the Pj are limit arithmetic subprogressions of [N] of
unbounded length (with the map j → Pj being a limit function), such that
n → e(αn) fluctuates by o(1) on each Pj (uniformly in j), and |E| = o(N).
Exercise 1.2.5. Use the previous three exercises to reprove Roth’s theorem.
Exercise 1.2.6 (Roth’s theorem in bounded characteristic). Let F be a
finite field, let δ 0, and let V be a finite vector space. Show that if
the dimension of V is suﬃciently large depending on F, δ, and if A ⊂ V
is such that |A| ≥ δ|V |, then there exists a, r ∈ V with r = 0 such that
a, a + r, a + 2r ∈ A. (Hint: Mimic the above arguments (either finitarily, or
with ultralimits), using hyperplanes as a substitute for subprogressions.)
Exercise 1.2.7 (Roth’s theorem in finite abelian groups). Let G be a finite
abelian group, and let δ 0. Show that if |G| is suﬃciently large depending
on δ, and A ⊂ G is such that |A| ≥ δ|G|, then there exists a, r ∈ V with
r = 0 such that a, a + r, a + 2r ∈ A. (Hint: If there is an element of
G of large order, one can use Theorem 1.2.2 and the pigeonhole principle.
If all elements have bounded order, one can instead use Exercise 1.2.6.)
This result (as well as the special case in the preceding exercise) was first
established by Meshulam [Me1995].
1.2.2. The energy increment argument. Now we turn to the energy
increment approach. This approach requires a bit more machinery to set
up, but ends up being quite flexible and powerful (for instance, it is the
starting point for my theorem with Ben Green establishing arbitrarily long
progressions in the primes, which we do not know how to establish via
density increment arguments).
Instead of passing from [N] to a subprogression, we now instead coarsen
[N] to some partition (or factor) of [N], as follows. Define a factor of [N]
to be a σ-algebra of subsets B of [N], or equivalently a partition of [N]
into disjoint atoms or cells (with the elements of B then being the arbitary
unions of atoms). Given a function f : [N] → C and a factor B, we define
the conditional expectation E(f|B): [N] → C to be the function whose value
at a given point x ∈ [N] is given by the formula