Transcript Alison Lewko's first talk

On sets of large doubling, ¤(4)
sets, and error-correcting codes
Allison Lewko
Columbia
University
Mark Lewko
Institute for
Advanced Study
Doubling of Sets
Sets of Large Doubling
A First Attempt at a Structure Theorem
Connection to ¤(4) Sets
Connection to ¤(4) Sets
A Question of Rudin
[Rudin, 1960]
Is every ¤(4) set a
finite union of B2[G]
sets?
Meyer’s Set [M68]
Ramsey’s Theorem
(1, 2)
(1, 13)
(2, 5)
(5, 12)
(13, 33)
(23, 42)
(8,10)
Meyer’s Set (contd.)
(2, 5)
(23, 42)
(1, 13)
(1, 2)
(5, 12)
...
Meyer’s Set (contd.)
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)
(1, 2, 4, 13)
(2, 5, 6, 10)
(5, 12, 24, 73)
(3, 11, 13, 33)
(7, 19, 23, 42)
(4, 8, 9, 10)
Properties of the Construction
Refining the Approach
a
+b
Reed-Solomon Codes
B2[1] Set Building Blocks
...
B2[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 B2[G]
sets?
Thanks!
Questions?