The \(m\)th real Johnson–Wilson spectrum \(ER(m)\) is a \(2^{m+2}(2^m-1)\)-periodic spectrum constructed from the \(2(2^m-1)\)-periodic spectrum \(E(m)\); for example \(ER(1)\) and \(E(1)\) are orthogonal and unitary \(K\)-theory localised at \(2\). The authors set up a Bockstein spectral sequence to compute \(ER(m)^*(X)\) from \(E(m)^*(X)\), and they compute \(ER(m)^*(X)\) when \(X\) is an infinite-dimensional real projective space. They then compute \(ER(2)^*(X)\), where \(X\) is a finite-dimensional real projective space \(\mathbb RP^n\). From these computations they obtain new nonimmersion theorems for \(\mathbb RP^n\). The lowest-dimensional new result is for \(\mathbb{R} P^{48}\).