Integral kašin splittings

For x ∈ ℝⁿ and p ≥ 1 put ∥x∥ := (n ^-1 ∑ |x_i|^p)^1/p. An orthogonal direct sum decomposition ℝ^2k = E ⊕ E⊥ where dim E = k and sup_{0≠x∈E∪E⊥} ∥x∥ ₂/∥x∥₁ ≤ C called here a (k, C)-splitting. By a theorem of Kašin 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 = 32eπ. 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^1/2π ^-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 of (k, C)-splittings.