On sets of large doubling,

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On sets of large doubling, (4) sets, and
error-correcting codes
Allison Lewko
Mark Lewko
Columbia University
Institute for Advanced Study
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Doubling of Sets
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Sets of Large Doubling
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A First Attempt at a Structure Theorem
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Connection to (4) Sets
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Connection to (4) Sets
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A Question of Rudin
Rudin, 1960 Is every (4) set a finite union of
B2G sets?
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Meyers Set M68
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Ramseys Theorem
(1, 2)
(1, 13)
(2, 5)
(13, 33)
(5, 12)
(23, 42)
(8,10)
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Meyers Set (contd.)
(23, 42)
(1, 13)
(2, 5)
(1, 2)
(5, 12)
. . .
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Meyers Set (contd.)
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Meyers Set (contd.)
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Third Attempt at a Structure Theorem
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Some Related Questions
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Attacking the Incompressible Union Problem
. . .
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Properties of This Construction
a
b
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Properties of This Construction
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Recall Ramseys Theorem
(1, 2, 3, 4)
(1, 2, 4, 13)
(2, 5, 6, 10)
(3, 11, 13, 33)
(5, 12, 24, 73)
(7, 19, 23, 42)
(4, 8, 9, 10)
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Properties of the Construction
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Refining the Approach
a
b
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Reed-Solomon Codes
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B21 Set Building Blocks
. . .
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B21 Set Building Blocks
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Assembling the Blocks
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Hadamard Matrices
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Summary of the Construction
. . .
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Proof of Incompressibility
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Implications of Construction
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Open Problems
Is every Sidon set a finite union of independent
sets?
What about a structure theorem for large doubling
sets by moving beyond B2G sets?
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Thanks!
Questions?