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The approximation constantλ  k    (  ζ  )is defined as the supremum of    η  ∈  R    such that the estimatemax   1  ⩽  j  ⩽  k∥    ζ   j   x  ∥   ⩽   x   ‐  ηhas infinitely many integer solutions   x  . Here    ∥  .  ∥    denotes the distance to the closest integer. We establish a connection on the joint spectrum    (   λ  1    (  ζ  )   ,   λ  2    (  ζ  )   ,  …  )   , which will lead to various improvements of known results on the individual spectrum of the approximation constantsλ  k    (  ζ  )as well. In particular, for given    k  ⩾  1    and    λ  ⩾  1   , we construct   ζ   in the Cantor set withλ  k    (  ζ  )   =  λ   . Moreover, we establish an estimate for the uniform approximation constantsλ  ^   k    (  ζ  )    , which enables us to determine classical approximation constants for Liouville numbers.

ON THE SPECTRUM OF DIOPHANTINE APPROXIMATION CONSTANTS