Gaps in bounded query hierarchies

Prior results show that most bounded query hierarchies cannot contain finite gaps. For example, it is known that P_(m+1)-tt ^SAT=P_m-tt^SAT⇒P_btt ^SAT=P_m-tt^SAT and for all sets A·FP_(m=1)-tt^A=FP_m-tt^A ⇒FP_btt^A=FP_m-tt^A ·P_(m+1)-T^A=P_m-T^A =P_bT^A·FP_(m+1)-T^A =FP_m-T^A⇒FP_bT^A=FP _m-T^A where P_m-tt^A is the set of languages computable by polynomial-time Turing machines that make m nonadaptive queries to A; P_btt^A=∪_mP _m-tt^A, P_m-t^A and P_bT ^A are the analogous adaptive queries classes; and FP _m-tt^A, FP_btt^A, FP_m-T ^A, and FP_bT^A in turn are the analogous function classes. It was widely expected that these general results would extend to the remaining case-languages computed with nonadaptive queries-yet results remained elusive. The best known was that P_2m-tt^A=P_m-tt^A⇒P _btt^A=P_m-tt^A. We disprove the conjecture, in fact, P_[4/3m]-tt^A=P_m-tt ^Anot⇒P_{([4/3m]+1)-tt}=P_[4/3m]-tt ^A. Thus there is a P_m-tt^A hierarchy that contains a finite gap. We also make progress on the 3-tt vs. 2-tt case: P_3-tt^A=P_2-tt^A⇒P _btt^A⊆P_2-tt^A/poly