Simultaneous Diophantine
Approximation with Excluded Primes
László Babai
Daniel Štefankovič
Dirichlet (1842)
Simultaneous Diophantine Approximation
Given reals

1 ,2 ,...,n , Q
r1 ,..., rn and q
such that q  Q and
integers
i q  ri  Q
iQ  pi  1/ 2
trivial
1/ n
for all
i
Simultaneous Diophantine Approximation
with an excluded prime
Given reals
?
1,2 ,...,n
integers
such that
r1 ,..., rn
prime
and
gcd(p, q)  1
qi  ri  
for all
q
and
i
p
Simultaneous diophantine
excluding p
 -approximation
Not always possible
Example
p3
If
1  1/ 3
then
 | q1  r1 || q / 3  r1 | 1/ 3
Simultaneous diophantine
excluding p
 -approximation
obstacle with 2 variables
If
1  22  1/ p
then
 | q1  r1 |
 | q2  r2 |
3 | q(1  22 )  (r1  2r2 ) | 1/ p
Simultaneous diophantine
excluding p
 -approximation
general obstacle
If
b11  b22  ...  bnn  1/ p  t
then
 | bi |  1/ p
Simultaneous diophantine
excluding p
 -approximation
Theorem:
If there is no  -approximation
excluding p then there exists an
obstacle with
| b |  n
3/ 2
i
/
Kronecker’s theorem ():
Arbitrarily good approximation excluding
possible IFF no obstacle.
p
Simultaneous diophantine
excluding p
 -approximation
obstacle with
| b |  n
3/ 2
i
/
necessary to prevent  -approximation
excluding p
sufficient to prevent

-approximation
pn
excluding p
3/ 2
Motivating example
Shrinking by stretching
Motivating example
set
A   Z / mZ
arc length of A
max | a (mod m) |
aA
stretching by
x
a
Ax  {ax | a  A}
gcd( x, m)  1
ax mod m
Example of the motivating example
A = 11-th roots of unity mod 11177
Example of the motivating example
A = 11-th roots of unity mod 11177
168
Shrinking modulo a prime
If m a prime
then
every small set can be shrunk
Shrinking modulo a prime
m a prime
d | A |
there exists x such that
11/ d
arc-length of Ax  m
proof:
a1
ad
,...,
m
m
Q : m  1
x : q
Dirichlet

q; 0  q  Q
1
qi  pi  1/ n
Q
Shrinking modulo any number
m
a prime
every small set can
be shrunk
?
Shrinking modulo any number
m
every small set can
be shrunk
a prime
m2
k 1
A  {1,1  2 }
k
If
gcd( x, m)  1
then the arc-length of
2
k 2
Ax
Where does the proof break?
m2
k
proof:
a1
ad
,...,
m
m
Q : m  1
x : q
Dirichlet

q; 0  q  Q
1
qi  pi  1/ n
Q
Where does the proof break?
m2
k
need:
approximation excluding 2
proof:
a1
ad
,...,
m
m
Q : m  1
x : q
Dirichlet

q; 0  q  Q
1
qi  pi  1/ n
Q
Shrinking cyclotomic classes
m
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
k
•Warring problem mod p
•intersection conditions modulo pk
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
excluding p then there exists an
obstacle with
| b |  n
i
3/ 2
/
Lattice
v1 ,..., vn  R
n
linearly independent
v1
v2
Lattice
v1 ,..., vn  R
n
v1Z  ...  vn Z
Lattice
v1 ,..., vn  R
n
v1Z  ...  vn Z
Dual lattice
L  {u |(v  L)v u  Z}
*
T
Banasczyk’s technique (1992)
gaussian weight of a set
 ( A)   e
 ||x||2
xA
mass displacement function of lattice
L ( x)   ( L  x) /  ( L)
Banasczyk’s technique (1992)
mass displacement function of lattice
L ( x)   ( L  x) /  ( L)
properties:
0  L ( x)  1
dist( x, L)  n
  L ( x )  1/ 4
Banasczyk’s technique (1992)
discrete measure
 L ( A)   ( L  A) /  ( L)
relationship between the discrete measure and
the mass displacement function of the dual
 L ( x )  L ( x)
*
1
2
T
 L ( x) 
exp(


||
y
||
)
exp(2

iy
x)

 ( L) yL
Banasczyk’s technique (1992)
discrete measure defined by the lattice
 L ( A)   ( L  A) /  ( L)
 L ( x )  L ( x)
*
1
*

 ( L) ||x|| s
1
*

 ( L) ||x|| s
1
2
T
 L ( x) 
exp(


||
y
||
)
exp(2

iy
x)

 ( L) yL
Banasczyk’s technique (1992)
1 ,2 ,3
1
0

0

0
0 0 1 

1 0 2 
n /
0 1 3 

0 0  
there is no short vector w  L
with coefficient of the
last column  0(mod p )
Banasczyk’s technique (1992)
there is no short vector w  L
with coefficient of the
last column  0(mod p )
 L (u)  1/ 2

u :
en 1
p n
 L (u )  1/ 2
*
dist(u, L )  n
*
obstacle
QED
Lovász (1982)
Simultaneous Diophantine Approximation
Given rationals
1 ,2 ,...,n , Q
can find in polynomial time
integers
0<q  Q
p1 ,..., pn
n2
2
qi  pi  1/ n
Q
for all
i
Factoring polynomials with rational coefficients.

Simultaneous diophantine -approximation
excluding p - algorithmic
Given rationals
1,2 ,...,n
,prime
can find in polynomial time
2Cn 1 p -approximation excluding p
where  is smallest such that there
exists  -approximation excluding p
Cn  4 n 2
n/2
p
Exluding prime and bounding denominator
If there is no  -approximation
excluding p with q  Q
then there exists an
approximate obstacle with
| b |  n
i
3/ 2
/
b11  b22  ...  bnn  1/ p  t  
|  | n / Q
Exluding prime and bounding denominator
the obstacle
necessary to prevent  -approximation
excluding p with q  Q
sufficient to prevent
3/ 2
 /(2n p) -approximation
excluding p with q  Q /(2 pn )
Exluding several primes
If there is no  -approximation
excluding p1 ,..., pk
then there exists
obstacle with
| b |  n
1/ 2
i
n
b
i 1
i
i


jA[ k ]
(max(n, k )) / 
1/ p j  t
Show that there is no small obstacle!
k
m=7
*
m
 primitive 3-rd root of unity
know
1      0 (mod 7 )
2
k
obstacle
c0  c1  t 7 , gcd(t ,7)  1
k 1
Show that there is no small obstacle!
1      0 (mod 7 )
k 1
c0  c1  t 7 , gcd(t ,7)  1
2
k
Res(1  x  x , c0  c1 x )
2
0
divisible by 7 k 1
 2( c02  c12 )

4
7
( k 1) / 2
There is g with all
3-rd roots
[(4 7)m1/ 2 ,(4 7)m1/ 2 ]
Dual lattice
 1
 0

 0

  1
 
0
0
1
0
0
1
2


3


0

0
0  / n

1


Algebraic integers?
possible that a small integer
combination with small coefficients
is doubly exponentially close to 1/p