A note on higher topological Hochschild homology

Chromatic redshift phenomena suggest that algebraic K-theory increases the height of a commutative ring spectrum by one. In many cases, the chromatic redshift is already detected by negative topological cyclic homology. This paper explores higher chromatic redshift via homotopy fixed points spectrum of higher topological Hochschild homology. Specifically, starting from a commutative ring spectrum that detects $v_n$-elements, the homotopy fixed points spectrum of higher topological Hochschild homology of it detects $v_{n+k}$-elements, with $k$ greater than one.