## Abstract

For x ∈ ℝ^{n} 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^{1/2}π ^{-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.

Original language | English (US) |
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Pages (from-to) | 139-156 |

Number of pages | 18 |

Journal | Israel Journal of Mathematics |

Volume | 138 |

DOIs | |

State | Published - Jan 1 2003 |