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In the previous set of notes, we saw that one could derive expansion of Cayley graphs from three ingredients: non-concentration, product theorems, and quasirandomness. Quasirandomness was discussed in Notes 3. In the current set of notes, we discuss product theorems. Roughly speaking, these theorems assert that in certain circumstances, a finite subset {A} of a group {G} either exhibits expansion (in the sense that {A^3}, say, is significantly larger than {A}), or is somehow “close to” or “trapped” by a genuine group.

Theorem 1 (Product theorem in {SL_d(k)}) Let {d \geq 2}, let {k} be a finite field, and let {A} be a finite subset of {G := SL_d(k)}. Let {\epsilon >0} be sufficiently small depending on {d}. Then at least one of the following statements holds:

  • (Expansion) One has {|A^3| \geq |A|^{1+\epsilon}}.
  • (Close to {G}) One has {|A| \geq |G|^{1-O_d(\epsilon)}}.
  • (Trapping) {A} is contained in a proper subgroup of {G}.

We will prove this theorem (which was proven first in the {d=2,3} cases for fields {F} of prime order by Helfgott, and then for {d=2} and general {F} by Dinai, and finally to general {d} and {F} independently by Pyber-Szabo and by Breuillard-Green-Tao) later in this notes. A more qualitative version of this proposition was also previously obtained by Hrushovski. There are also generalisations of the product theorem of importance to number theory, in which the field {k} is replaced by a cyclic ring {{\bf Z}/q{\bf Z}} (with {q} not necessarily prime); this was achieved first for {d=2} and {q} square-free by Bourgain, Gamburd, and Sarnak, by Varju for general {d} and {q} square-free, and finally by this paper of Bourgain and Varju for arbitrary {d} and {q}.

Exercise 1 (Girth bound) Assuming Theorem 1, show that whenever {S} is a symmetric set of generators of {SL_d(k)} for some finite field {k} and some {d\geq 2}, then any element of {SL_d(k)} can be expressed as the product of {O_d( \log^{O_d(1)} |k| )} elements from {S}. (Equivalently, if we add the identity element to {S}, then {S^m = SL_d(k)} for some {m = O_d( \log^{O_d(1)} |k| )}.) This is a special case of a conjecture of Babai and Seress, who conjectured that the bound should hold uniformly for all finite simple groups (in particular, the implied constants here should not actually depend on {d}. The methods used to handle the {SL_d} case can handle other finite groups of Lie type of bounded rank, but at present we do not have bounds that are independent of the rank. On the other hand, a recent paper of Helfgott and Seress has almost resolved the conjecture for the permutation groups {A_n}.

A key tool to establish product theorems is an argument which is sometimes referred to as the pivot argument. To illustrate this argument, let us first discuss a much simpler (and older) theorem, essentially due to Freiman, which has a much weaker conclusion but is valid in any group {G}:

Theorem 2 (Baby product theorem) Let {G} be a group, and let {A} be a finite non-empty subset of {G}. Then one of the following statements hold:

  • (Expansion) One has {|A^{-1} A| \geq \frac{3}{2} |A|}.
  • (Close to a subgroup) {A} is contained in a left-coset of a group {H} with {|H| < \frac{3}{2} |A|}.

To prove this theorem, we suppose that the first conclusion does not hold, thus {|A^{-1} A| <\frac{3}{2} |A|}. Our task is then to place {A} inside the left-coset of a fairly small group {H}.

To do this, we take a group element {g \in G}, and consider the intersection {A\cap gA}. A priori, the size of this set could range from anywhere from {0} to {|A|}. However, we can use the hypothesis {|A^{-1} A| < \frac{3}{2} |A|} to obtain an important dichotomy, reminiscent of the classical fact that two cosets {gH, hH} of a subgroup {H} of {G} are either identical or disjoint:

Proposition 3 (Dichotomy) If {g \in G}, then exactly one of the following occurs:

  • (Non-involved case) {A \cap gA} is empty.
  • (Involved case) {|A \cap gA| > \frac{|A|}{2}}.

Proof: Suppose we are not in the pivot case, so that {A \cap gA} is non-empty. Let {a} be an element of {A \cap gA}, then {a} and {g^{-1} a} both lie in {A}. The sets {A^{-1} a} and {A^{-1} g^{-1} a} then both lie in {A^{-1} A}. As these sets have cardinality {|A|} and lie in {A^{-1}A}, which has cardinality less than {\frac{3}{2}|A|}, we conclude from the inclusion-exclusion formula that

\displaystyle |A^{-1} a \cap A^{-1} g^{-1} a| > \frac{|A|}{2}.

But the left-hand side is equal to {|A \cap gA|}, and the claim follows. \Box

The above proposition provides a clear separation between two types of elements {g \in G}: the “non-involved” elements, which have nothing to do with {A} (in the sense that {A \cap gA = \emptyset}, and the “involved” elements, which have a lot to do with {A} (in the sense that {|A \cap gA| > |A|/2}. The key point is that there is a significant “gap” between the non-involved and involved elements; there are no elements that are only “slightly involved”, in that {A} and {gA} intersect a little but not a lot. It is this gap that will allow us to upgrade approximate structure to exact structure. Namely,

Proposition 4 The set {H} of involved elements is a finite group, and is equal to {A A^{-1}}.

Proof: It is clear that the identity element {1} is involved, and that if {g} is involved then so is {g^{-1}} (since {A \cap g^{-1} A = g^{-1}(A \cap gA)}. Now suppose that {g, h} are both involved. Then {A \cap gA} and {A\cap hA} have cardinality greater than {|A|/2} and are both subsets of {A}, and so have non-empty intersection. In particular, {gA \cap hA} is non-empty, and so {A \cap g^{-1} hA} is non-empty. By Proposition 3, this makes {g^{-1} h} involved. It is then clear that {H} is a group.

If {g \in A A^{-1}}, then {A \cap gA} is non-empty, and so from Proposition 3 {g} is involved. Conversely, if {g} is involved, then {g \in A A^{-1}}. Thus we have {H = A A^{-1}} as claimed. In particular, {H} is finite. \Box

Now we can quickly wrap up the proof of Theorem 2. By construction, {A \cap gA| > |A|/2} for all {g \in H},which by double counting shows that {|H| < 2|A|}. As {H = A A^{-1}}, we see that {A} is contained in a right coset {Hg} of {H}; setting {H' := g^{-1} H g}, we conclude that {A} is contained in a left coset {gH'} of {H'}. {H'} is a conjugate of {H}, and so {|H'| < 2|A|}. If {h \in H'}, then {A} and {Ah} both lie in {H'} and have cardinality {|A|}, so must overlap; and so {h \in A A^{-1}}. Thus {A A^{-1} = H'}, and so {|H'| < \frac{3}{2} |A|}, and Theorem 2 follows.

Exercise 2 Show that the constant {3/2} in Theorem 2 cannot be replaced by any larger constant.

Exercise 3 Let {A \subset G} be a finite non-empty set such that {|A^2| < 2|A|}. Show that {AA^{-1}=A^{-1} A}. (Hint: If {ab^{-1} \in A A^{-1}}, show that {ab^{-1} = c^{-1} d} for some {c,d \in A}.)

Exercise 4 Let {A \subset G} be a finite non-empty set such that {|A^2| < \frac{3}{2} |A|}. Show that there is a finite group {H} with {|H| < \frac{3}{2} |A|} and a group element {g \in G} such that {A \subset Hg \cap gH} and {H = A A^{-1}}.

Below the fold, we give further examples of the pivot argument in other group-like situations, including Theorem 2 and also the “sum-product theorem” of Bourgain-Katz-Tao and Bourgain-Glibichuk-Konyagin.

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