 
Summary: INTEGRAL KA SIN SPLITTINGS
GREG W. ANDERSON
Abstract. For x 2 R n and p 1 put kxk p := (n 1
P
jx i j p ) 1=p .
An orthogonal direct sum decomposition R 2k = E E ? where
dim E = k and sup 06=x2E[E ? kxk 2 = kxk 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 > 8e 1=2 1=2 and
positive integer k we specify a positive integer N(k; C) such that
in the set of k by 2k matrices with integral entries of absolute
value not exceeding N(k; C) there exists a matrix with row span
a summand in a (k; C)splitting. We have N(k; C) 2 18k for k
large enough depending on C. We explain in detail how to test a
matrix for the property of representing a (k; C)splitting. Taken
together our results yield an explicit (if impractical) construction
