The notion of the bounded jump was introduced by Anderson and Csima and used to study the effect $wtt$-functionals have on the Halting problem. They proved that the bounded jump shared similar properties with the (classical) Turing jump, but was more suitable when dealing with weak truth table degrees, and showed that Shoenfield's jump inversion holds for the bounded jump. Anderson, Csima and Lange then studied the corresponding notions of lowness and highness for the bounded jump - the bounded low and bounded high sets, and showed that these notions do not line up well with the classical notions of lowness and highness.

We show that Sacks' jump inversion fails for the bounded jump, and characterize the bounded high c.e. sets. We also study the relationship the bounded high weak truth table degrees have with different notions of effective domination.