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1. Introduction. The basic object of Diophantine approximation is rational approximation to points u in R n . This is generally measured by elements of the extended real line [−∞, ∞] called exponents of approximation to u. The spectrum of a family of exponents (µ 1 , . . . , µ m ) is the subset of [−∞, ∞] m consisting of all m-tuples (µ 1 (u), . . . , µ m (u)) as u varies among the points of R n with linearly independent coordinates over Q. In all cases where such a spectrum has been explicitlydoi:10.4064/aa191217-9-4 fatcat:mpwrqcgcgbhnhatna63s23ov2a