On sets of large doubling, ¤ (4) sets, and error-correcting codes Allison LewkoMark Lewko Columbia University Institute for Advanced Study TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AAA

Doubling of Sets

Sets of Large Doubling

A First Attempt at a Structure Theorem

Connection to ¤ (4) Sets

A Question of Rudin [Rudin, 1960] Is every ¤ (4) set a finite union of B 2 [G] sets?

Meyer’s Set [M68]

Ramsey’s Theorem (1, 2) (2, 5) (1, 13) (23, 42) (13, 33) (8,10) (5, 12)

Meyer’s Set (contd.)... (1, 2) (2, 5) (1, 13) (23, 42) (5, 12)

Meyer’s Set (contd.)

Third Attempt at a Structure Theorem

Some Related Questions

Attacking the “Incompressible Union” Problem...

Properties of This Construction a + b

Properties of This Construction

Recall: Ramsey’s Theorem (1, 2, 3, 4) (2, 5, 6, 10) (1, 2, 4, 13) (7, 19, 23, 42) (3, 11, 13, 33) (4, 8, 9, 10) (5, 12, 24, 73)

Properties of the Construction

Refining the Approach a + b

Reed-Solomon Codes

B 2 [1] Set Building Blocks...

B 2 [1] Set Building Blocks

Assembling the Blocks

Hadamard Matrices

Summary of the Construction...

Proof of “Incompressibility”

Implications of Construction

Open Problems Is every Sidon set a finite union of independent sets? What about a structure theorem for large doubling sets by moving beyond B 2 [G] sets?

Thanks! Questions?