Simultaneous Diophantine Approximation with Excluded Primes

1. Simultaneous Diophantine Approximation with Excluded Primes László Babai Daniel Štefankovič

2. Dirichlet (1842) Simultaneous Diophantine Approximation Given reals and integers such that and for all trivial

3. Simultaneous Diophantine Approximation with an excluded prime Given reals prime ? and integers and such that for all

4. Simultaneous diophantine -approximation excluding Not always possible Example If then

5. Simultaneous diophantine -approximation excluding obstacle with 2 variables If then

6. Simultaneous diophantine -approximation excluding general obstacle If then

7. Simultaneous diophantine -approximation excluding Theorem: If there is no -approximation excluding then there exists an obstacle with Kronecker’s theorem (): Arbitrarily good approximation excluding possible IFF no obstacle.

8. Simultaneous diophantine -approximation excluding obstacle with necessary to prevent -approximation excluding sufficient to prevent -approximation excluding

9. Motivating example Shrinking by stretching

10. Motivating example set arc length of A stretching by

11. Example of the motivating example A = 11-th roots of unity mod 11177

12. Example of the motivating example A = 11-th roots of unity mod 11177 168

13. Shrinking modulo a prime a prime If then every small set can be shrunk

14. Shrinking modulo a prime a prime there exists such that arc-length of proof: Dirichlet

15. Shrinking modulo any number every small set can be shrunk a prime ?

16. Shrinking modulo any number every small set can be shrunk a prime If then the arc-length of

17. Where does the proof break? proof: Dirichlet

18. Where does the proof break? need: approximation excluding 2 proof: Dirichlet

19. Shrinking cyclotomic classes every small set can be shrunk a prime set of interest – cyclotomic class (i.e. the set of r-th roots of unity mod m) • locally testable codes • diameter of Cayley graphs • Warring problem mod p • intersection conditions modulo p k k

20. Shrinking cyclotomic classes cyclotomic class can be shrunk

21. Shrinking cyclotomic classes cyclotomic class can be shrunk Show that there is no small obstacle!

22. Theorem: If there is no -approximation excluding then there exists an obstacle with

23. Lattice linearly independent

24. Lattice

25. Lattice Dual lattice

26. Banasczyk’s technique (1992) gaussian weight of a set mass displacement function of lattice

27. Banasczyk’s technique (1992) mass displacement function of lattice properties:

28. Banasczyk’s technique (1992) discrete measure relationship between the discrete measure and the mass displacement function of the dual

29. Banasczyk’s technique (1992) discrete measure defined by the lattice

30. Banasczyk’s technique (1992) there is no short vector with coefficient of the last column

31. Banasczyk’s technique (1992) there is no short vector with coefficient of the last column obstacle QED

32. Lovász (1982) Simultaneous Diophantine Approximation Given rationals can find in polynomial time integers for all Factoring polynomials with rational coefficients.

33. Simultaneous diophantine -approximation excluding - algorithmic Given rationals ,prime can find in polynomial time -approximation excluding where is smallest such that there exists -approximation excluding

35. Exluding prime and bounding denominator If there is no -approximation excluding with then there exists an approximate obstacle with

36. Exluding prime and bounding denominator the obstacle necessary to prevent -approximation excluding with sufficient to prevent -approximation excluding with

37. Exluding several primes If there is no -approximation excluding then there exists obstacle with

38. Show that there is no small obstacle! k *  m=7 m  primitive 3-rd root of unity know obstacle

39. Show that there is no small obstacle! divisible by There is g with all 3-rd roots

40. Dual lattice

41. Algebraic integers? possible that a small integer combination with small coefficients is doubly exponentially close to 1/p