If the detector is stationary and oscillates at the same frequency as the electron, it would indeed detect the electromagnetic (EM) wave produced by the oscillating electron.
When an electron oscillates, it generates an electromagnetic wave that propagates outward from the source. Electromagnetic waves consist of oscillating electric and magnetic fields, and they can be detected by appropriately designed detectors.
If the detector oscillates with the same frequency as the electron, it will effectively remain in sync with the oscillating electric and magnetic fields of the electromagnetic wave. As a result, the detector will detect the presence of the electromagnetic wave and register the associated energy and intensity.
This phenomenon is analogous to the concept of resonance, where systems oscillate at the same frequency, enhancing the response. In this case, the detector is effectively resonating with the oscillating electron and can detect the EM wave produced by it.
It's important to note that achieving such perfect synchronization between the detector and the electron's oscillation may be challenging in practice, and it depends on the specific experimental setup and technical limitations.