ON THE WARING-GOLDBACH PROBLEM WITH ALMOST EQUAL SUMMANDS

We use transference principle to show that whenever s is suitably large depending on k2, every sufficiently large natural number n satisfying some congruence conditions can be written in the form n=p1k</mml:msubsup>++psk</mml:msubsup>, where <mml:msub>p1,...,<mml:msub>ps is an element of [x-x theta ,x+x theta] are primes, x=(n/s)1/k and theta =0.525+epsilon. We also improve known results for theta when k2 and sk2+k+1. For example, when k4 and sk2+k+1 we have theta =0.55+epsilon. All previously known results on the problem had theta >3/4.