1. 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

2. Doubling of Sets

3. Sets of Large Doubling

4. A First Attempt at a Structure Theorem

5. Connection to ¤ (4) Sets

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

8. Meyer’s Set [M68]

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

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

11. Meyer’s Set (contd.)

13. Third Attempt at a Structure Theorem

14. Some Related Questions

15. Attacking the “Incompressible Union” Problem...

16. Properties of This Construction a + b

17. Properties of This Construction

18. 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)

19. Properties of the Construction

20. Refining the Approach a + b

21. Reed-Solomon Codes

22. B 2 [1] Set Building Blocks...

23. B 2 [1] Set Building Blocks

24. Assembling the Blocks

25. Hadamard Matrices

26. Summary of the Construction...

27. Proof of “Incompressibility”

28. Implications of Construction

29. 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?

30. Thanks! Questions?