A comprehensive search for long and short periodic features from an extremely active cycle of FRB 20240114A

Possible periodic features in fast radio bursts (FRBs) may provide insights into their astrophysical origins. Using extensive observations from the five-hundred-meter aperture spherical radio telescope (FAST), we conduct a multi-timescale periodicity search for the exceptionally active repeater FRB 20240114A. Our analysis is based on different datasets for different timescales: for short-timescale periodicity in time of arrivals (TOAs), we use 57 observations from January to August 2024; for long-timescale periodicity, we employ an extended TOA dataset comprising 111 observations spanning from January 2024 to October 2025; and for burst time series analysis, we utilize individual burst data from the 57 FAST observations. We identify three candidate short-timescale periodic signals (0.673, 0.635, and 0.536 s) with significances of $3.2\sigma$-$6\sigma$, each detected in two independent observations. On longer timescales, we detect a significant ($143.40\pm7.19$)-d periodicity with $5.2\sigma$ significance, establishing FRB 20240114A as a periodic repeater. In burst time series, we find quasi-periodic oscillations in the few hundred Hz range ($3.4\sigma$ and $3.7\sigma$) and periodic burst trains with periods of several to tens of milliseconds (3$\sigma$-3.9$\sigma$), though these periodic features appear transient and short-lived. The detection of periodic signals at these different timescales indicates that FRB 20240114A exhibits intriguing periodic self-similar characteristics. Despite the comprehensive dataset, no definitive periodicity linked to the source's rotation is confirmed, placing stringent constraints on the intrinsic source properties and the modulation mechanisms. All data are available via the Science Data Bank.