This paper addresses the problem of activity recognition and people identification using accelerometer signals acquired by personal devices. Specifically, we propose a framework based on a Deep Neural Network that employs an efficient dense trajectory encoding to compute features. These Accelerometer Dense Trajectory (ADT) features, which are similar to those used for action recognition in the spatio-temporal domain of video data, densely map the accelerometer signals into three-dimensional normalised positions. To deal with the unordered nature and dimensional variation of trajectories associated with the classes, the proposed framework employs Fisher Vectors as a high order representation of the extracted features. We evaluate the proposed ADT features and framework on the Sphere2016 Challenge and WISDM datasets for activity recognition. For people identification, we employ the RecodGait dataset. For these two significantly different classification tasks, the performance evaluation results confirm the high descriptiveness of the proposed ADT features and the effectiveness of the proposed framework.