Simultaneous Diophantine Approximation with Excluded Primes László Babai Daniel Štefankovič

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

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

Simultaneous diophantine -approximation excluding Not always possible Example If then

Simultaneous diophantine -approximation excluding obstacle with 2 variables If then

Simultaneous diophantine -approximation excluding general obstacle If then

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

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

Motivating example Shrinking by stretching

Motivating example set stretching by arc length of A

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

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

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

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

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

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

proof: Dirichlet Where does the proof break?

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

Shrinking cyclotomic classes a prime every small set can be shrunk 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

Shrinking cyclotomic classes cyclotomic class can be shrunk

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

Theorem: If there is no-approximation excludingthen there exists an obstacle with

Lattice linearly independent

Lattice

Dual lattice

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

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

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

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

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

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

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

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

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

Exluding prime and bounding denominator necessary to prevent-approximation excludingwith the obstacle sufficient to prevent -approximation excludingwith

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

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

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

Dual lattice

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