 
Summary: INTEGRAL KASIN SPLITTINGS
GREG W. ANDERSON
Abstract. For x Rn
and p 1 put x p := (n1
xip
)1/p
.
An orthogonal direct sum decomposition R2k
= E E
where
dim E = k and sup0=xEE x 2 / x 1 C is called here
a (k, C)splitting. By a theorem of Kasin there exists C > 0
such that (k, C)splittings exist for all k, and by the volume ratio
method of Szarek one can take C = 32e. All proofs of existence
of (k, C)splittings heretofore given are nonconstructive.
Here we investigate the representation of (k, C)splittings by
matrices with integral entries. For every C > 8e1/2
1/2
and
positive integer k we specify a positive integer N(k, C) such that
