Rearrangements of functions on totally disconnected groups

A result of B.B. Wells Jr. claims that every complex valued continuous function on the compact ring Zp of p-adic integers has a rearrangement which belongs to a certain class (W) of functions having absolutely convergent Fourier series. We point out that this is not the case since every real valued function from (W) has Lebesgue null range. On the other hand we prove the existence of a rearrangement with absolutely convergent Fourier series for every continuous real valued function on Zp and on some other compact metric totally disconnected Abelian groups. We leave open if the same holds for all continuous complex valued functions on Zp.  2003 Elsevier Inc. All rights reserved