Sums, Products, and Rectangles

July 11, 2012

Joint work in progress with Victor Lie, Princeton.

Table of contents 1 Sums and Products 2 3 4

Preliminary definitions We consider sets of real numbers. Now: A is a *finite* set Later: A is a δ-neighborhood of a finite set Definition A + A = {a + b : a, b A} AA = {ab : a, b A}

Basic Observations A A + A A 2 A + A is (essentially) minimized when A is (essentially) an arithmetic progression

Basic Observations A A + A A 2 A + A is (essentially) minimized when A is (essentially) an arithmetic progression A AA A 2 AA is (essentially) minimized when A is (essentially) a geometric progression

A more precise statement Theorem (Erdös-Szemeredi, Nathanson,..., Elekes, Solymosi,...?? ) If A R, then World Record β = 3 11 A + A + AA A 1+β.

A more precise statement Theorem (Erdös-Szemeredi, Nathanson,..., Elekes, Solymosi,...?? ) If A R, then A + A + AA A 1+β. World Record β = 3 11 Freiman-type theorems are not effective when the doubling is A A ɛ, so the heuristic on the previous page is difficult to use

A slightly different problem sum-product theorem A + AA is large

A slightly different problem sum-product theorem A + AA is large blah blah blah about passing to a large subset, etc. The formuation with A + AA is more closely tied to the part of the talk Definition A + AA = {a + bc : a, b, c A}

Encoding sums-products as points-lines Michael Bateman, UCLA Figure \ Cambridge : (a, b) Sums, Products, l and Given a pair (a, b) A A, we have a line l a,b given by y = ax + b. This gives us A A set of lines: L = {l a,b : a, b A}.

Incidence theory (Szemeredi-Trotter) = A + xa large for typical x A = A + AA large Figure : P = Black Red

Szemeredi-Trotter Let P be a set of points in the plane, and let L be a set of lines in the plane. Write I (P, L) to denote the number of incidences between P and L: I (P, L) = #{(p, l) P L: p l}. Theorem Suppose P = L = N. Then I (P, L) N 4 3.

2-D Kakeya R is a collection of δ 1 rectangles.

2-D Kakeya R is a collection of δ 1 rectangles. m R is the multiplicity function (DRAW PICTURE FOR AUDIENCE)

2-D Kakeya R is a collection of δ 1 rectangles. m R is the multiplicity function (DRAW PICTURE FOR AUDIENCE) Theorem {m R > λ} m R 2 2 λ 2 log #R m R 1 λ 2.

2-D Kakeya R is a collection of δ 1 rectangles. m R is the multiplicity function (DRAW PICTURE FOR AUDIENCE) Theorem {m R > λ} m R 2 2 λ 2 log #R m R 1 λ 2. L 2 controlled by L 1 means not too much pile-up

Sums and Products Radial example Figure : {mr > λ} = lower left corner

Where is m R large? Upper bound on {m R > λ} cannot be improved without further hypothesis

Where is m R large? Upper bound on {m R > λ} cannot be improved without further hypothesis If upper bound is sharp, what can be said about structure of {m R > λ}?

Where is m R large? Always Upper bound on {m R > λ} cannot be improved without further hypothesis If upper bound is sharp, what can be said about structure of {m R > λ}? {m R > λ} 1 λ 2 CONJECTURE If {m R > λ} is not essentially concentrated in a ball, then {m R > λ} 1 λ 2+ɛ

Figure : Cantor-Kakeya set

Digression on Motivation Bounds on the maximal Schoedinger operator can be reduced to geometric questions of this form (Geometric result implies Fourier analytic result)

Digression on Motivation Bounds on the maximal Schoedinger operator can be reduced to geometric questions of this form (Geometric result implies Fourier analytic result) Consider a set E containing an α-dimensional set in each direction. How big is dimension of E? (Furstenberg conjecture)

Why should the conjecture be true? Simple case: Suppose the set {m R > λ} looks like a lattice with substantial separation? Conjecture is true here Szemeredi-Trotter style arguments work

Toward sum-product Let P be the collection of δ δ squares forming {m R > λ}

Toward sum-product Let P be the collection of δ δ squares forming {m R > λ} These δ δ squares are now the points Incidence theorem for P, R implies estimate on {m R > λ}

Toward sum-product Let P be the collection of δ δ squares forming {m R > λ} These δ δ squares are now the points Incidence theorem for P, R implies estimate on {m R > λ} What can we say when points in P have some separation?

Simplifying assumptions What if we assume the rectangles arise from pairs (a, b) A A?

Simplifying assumptions What if we assume the rectangles arise from pairs (a, b) A A? What if we additionally assume each point in P projects to A?

Simplifying assumptions What if we assume the rectangles arise from pairs (a, b) A A? What if we additionally assume each point in P projects to A? Counting incidences is like estimating A + AA, or more precisely, E x A A + xa

These are rectangles this time Figure : Incidences and sum-product

Theorem (Bourgain) if A has dimension 1 2. In fact #(A + AA) is large #(A + A) + #(AA) (#A) 1+ɛ #(A + xa) is large for a typical x A Bourgan proves the theorem for α (0, 1), with ɛ = ɛ(α) α 2 Probably ɛ(α) α, which is the correct numerology.

Why can t we appeal to classical sum-product theory? A naive approach might be to model intervals by their centers C, apply sum-product theory to this set C, then unwind Figure : Few δ-intervals, many distinct points

Difference in incidence approach for lines... Figure : Two lines intersect in one point

... Sums and Products Figure : Transverse intersection

... and that for rectangles Figure : Non-transverse intersection

Why should Bourgain s theorem be true? Consider a further special case: A is a genuine arithmetic progression with substantial separation between the intervals Slopes of tubes come from A, hence if slopes are different then they are very different Separation of slopes is almost-transversality

Further comments Again, Freiman s theorem is too weak Bourgain uses a multiscale analysis of the set A, and applies Freiman at each level Key simplification for Bourgain: product structure of rectangles and points Without product structure: separation of points (which we assume) is essentially dual condition to separation of slopes (which we know is helpful)