Parameter definability in the recursively enumerable degrees

The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the Σk relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k ≥ 7, the Σk relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that Low1 is parameter definable, and we provide a new example of a ∅–definable ideal. Moreover, we prove that automorphisms restricted to intervals [d,1], d 6= 0, are Σ7. We also show that, for each c 6= 0, (N,+,×) can be interpreted in [0, c] without parameters.