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We obtain restriction estimates of  ε  -removal type for the set of k -th powers of integers, and for discrete d -dimensional surfaces of the form{   (   n  1   ,  ⋯  ,   n  d   ,   n  1  k   +  ⋯  +   n  d  k   ):    |   n  1   |  ,  ⋯  ,  |   n  d   |  ⩽  N  }  ,which we term ' k -paraboloids'. For these surfaces, we obtain a satisfying range of exponents for large values of d ,  k . We also obtain estimates of  ε  -removal type in the full supercritical range for k -th powers and for k -paraboloids of dimension   d  &lt;  k  (  k  -  2  )   . We rely on a variety of techniques in discrete harmonic analysis originating in Bourgain's works on the restriction theory of the squares and the discrete parabola.

Restriction estimates of $$\varepsilon $$ ε -removal type for k-th powers and paraboloids