Models of arithmetic and the 1-3-1 lattice

J. Paris

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Displaying similar documents to “Models of arithmetic and the 1-3-1 lattice”

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Let $q, q_{1}, \dots, q_{s} \geq 2$ be integers, and let $α_{1}, α_{2}, \dots$ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence ${(\{α_{m} q^{n}\}, \dots, \{α_{m + s - 1} q^{n}\})}_{m, n = 1}^{M N}$ coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences ${(x n)}_{n = 1}^{M N}$ in $s$ -dimensional unit cube $(s, M, N = 1, 2, \dots)$ . We also find a lower bound of the discrepancy (up to a logarithmic factor) of the sequence ${(\{α_{1} q_{1}^{n}\}, \dots, \{α_{s} q_{s}^{n}\})}_{n = 1}^{N}$ (Korobov’s problem).

Boolean algebra valued measures

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The E and K functionals for the pair (X (A), l(B)).

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We prove some exact formulas for the E and K functionals for pairs of the type (X(A),l sub ∞ (B)) where X has the lattice property. These formulas are extensions of their well-known counterparts in the scalar valued case. In particular we generalize formulas by Pisier and by the present author.