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We study the  C   p  ‐equivariant Tate construction on the topological Hochschild homology THH( B ) of a symmetric ring spectrum  B  by relating it to a topological version  R  + ( B ) of the Singer construction, extended by a natural circle action. This enables us to prove that the fixed and homotopy fixed point spectra of THH( B ) are  p ‐adically equivalent for  B =MU and BP. This generalizes the classical  C   p  ‐equivariant Segal conjecture, which corresponds to the case  B = S .

The Segal conjecture for topological Hochschild homology of complex cobordism