On the Waring–goldbach Problem With Almost Equal Summands

We use transference principle to show that whenever s is suitably large depending on k ⩾ 2 , every sufficiently large natural number n satisfying some congruence conditions can be written in the form n = p 1 k + ⋯ + p s k , where p 1 , … , p s ∈ [ x − x θ , x + x θ ] are primes, x = ( n / s ) 1 / k and θ = 0.525 + ε . We also improve known results for θ when k ⩾ 2 and s ⩾ k 2 + k + 1 . For example, when k ⩾ 4 and s ⩾ k 2 + k + 1 we have θ = 0.55 + ε . All previously known results on the problem had θ > 3 / 4 .