Simultaneous Diophantine

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Simultaneous Diophantine Approximation
with Excluded Primes
László Babai Daniel Štefankovic
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Dirichlet (1842) Simultaneous Diophantine
Approximation

Given reals
and
integers
such that and
for all
trivial
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Simultaneous Diophantine Approximation with an
excluded prime
Given reals
prime
?
and
integers
and
such that
for all
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Simultaneous diophantine -approximation excludi
ng
Not always possible
Example
If
then
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Simultaneous diophantine -approximation excludi
ng
obstacle with 2 variables
If
then
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Simultaneous diophantine -approximation excludi
ng
general obstacle
If
then
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Simultaneous diophantine -approximation excludi
ng
Theorem

If there is no
-approximation
excluding
then there exists an
obstacle with
Kroneckers theorem (?)
Arbitrarily good approximation excluding
possible IFF no obstacle.
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Simultaneous diophantine -approximation excludi
ng
obstacle with
necessary to prevent -approximation
excluding
sufficient to prevent -approximation
excluding
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Motivating example
Shrinking by stretching
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Motivating example
set
arc length of A
stretching by
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Example of the motivating example
A 11-th roots of unity mod 11177
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Example of the motivating example
A 11-th roots of unity mod 11177
168
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Shrinking modulo a prime
a prime
If
then
every small set can be shrunk
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Shrinking modulo a prime
a prime

there exists such that
arc-length of
proof
Dirichlet
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Shrinking modulo any number
every small set can be shrunk
a prime
?
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Shrinking modulo any number
every small set can be shrunk
a prime
If
then the arc-length of
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Where does the proof break?
proof
Dirichlet
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Where does the proof break?
need
approximation excluding 2
proof
Dirichlet
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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
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Shrinking cyclotomic classes
cyclotomic class
can be shrunk
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Shrinking cyclotomic classes
cyclotomic class
can be shrunk
Show that there is no small obstacle!
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Theorem

If there is no
-approximation
excluding
then there exists an
obstacle with
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Lattice
linearly independent
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Lattice
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Lattice
Dual lattice
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Banasczyks technique (1992)
gaussian weight of a set

mass displacement function of lattice
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Banasczyks technique (1992)
mass displacement function of lattice
properties


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Banasczyks technique (1992)
discrete measure
relationship between the discrete measure and
the mass displacement function of the dual

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Banasczyks technique (1992)
discrete measure defined by the lattice

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Banasczyks technique (1992)
there is no short vector with coefficient of
the last column
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Banasczyks technique (1992)
there is no short vector with coefficient of
the last column
obstacle
QED
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Lovász (1982) Simultaneous Diophantine
Approximation
Given rationals
can find in polynomial time
integers
for all
Factoring polynomials with rational coefficients.
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Simultaneous diophantine -approximation excludi
ng
- algorithmic
Given rationals
,prime
can find in polynomial time
-approximation excluding
where
is smallest such that there
exists
-approximation excluding
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(No Transcript)
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Exluding prime and bounding denominator
If there is no
-approximation
excluding
with
then there exists an
approximate obstacle with
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Exluding prime and bounding denominator
the obstacle
necessary to prevent
-approximation
excluding
with
sufficient to prevent
-approximation
excluding
with
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Exluding several primes
If there is no
-approximation
excluding
then there exists
obstacle with
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Show that there is no small obstacle!
k

?
m7
m
? primitive 3-rd root of unity
know
obstacle
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Show that there is no small obstacle!
divisible by
There is g with all 3-rd roots
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Dual lattice
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Algebraic integers?
possible that a small integer combination with
small coefficients is doubly exponentially close
to 1/p