We prove that the Pythagoras number of the ring of 0regulous functions R 0 (R n ) is finite and bounded from above by 2 n . We also show that a k-regulous function which is nonnegative on R n cannot be necessarily written as a sum of squares of k-regulous functions, provided that k > 0 and n > 1. We then obtain lower bounds for p(R k (R n )) of k-regulous functions on R n , which tends to infinity as k tends to infinity for a fixed n ≥ 3.

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We prove that a k-regulous function defined on a two-dimensional nonsingular affine variety can be extended to an ambient variety. Additionally we derive some results concerning sums of squares of k-regulous functions; in particular we show that every positive semi-definite regular function on a non-singular affine variety can be written as a sum of squares of locally Lipschitz regulous functions.

Let G be the group PAff+(S 1 ) of piecewise-affine circle homeomorphisms or the group Diff ∞ (R/Z) of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in G is given.