We will give an overview of four families of cardinal characteristics defined on subspaces ∏ α ∈ κ b ( α ) of the generalised Baire space κ κ , where κ is strongly inaccessible and b ∈ κ κ . The considered families are bounded versions of the dominating, eventual difference, localisation and antilocalisation numbers, and their dual cardinals. We investigate parameters for which these cardinals are nontrivial and how the cardinals relate to each other and to other cardinals of the generalised Cichoń diagram. Finally we prove that different choices of parameters may lead to consistently distinct cardinals.

Abstract Given a cofinal cardinal function $h\in {}^{\kappa }\kappa $ for $\kappa $ inaccessible, we consider the dominating h-localisation number, that is, the least cardinality of a dominating set of h-slaloms such that every $\kappa $ -real is localised by a slalom in the dominating set. It was proved in [3] that the dominating localisation numbers can be consistently different for two functions h (the identity function and the power function). We will construct a $\kappa ^+$ -sized family of functions h and their corresponding localisation numbers, and use a ${\leq }\kappa $ -supported product of a cofinality-preserving forcing to prove that any simultaneous assignment of these localisation numbers to cardinals above $\kappa $ is consistent. This answers an open question from [3].