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Compressive Sensing (CS) has well boosted massive random access protocols over the last decade. Usually, on physical layer, the protocols employ some fat matrix with the property that sparse vectors in the much larger column space domain can still be recovered. This, in turn, greatly reduces the chances of collisions between access devices. This basic scheme has meanwhile been enhanced in various directions but the system cannot operate in overload regime, i.e. sustain significantly more users than the row dimension of the fat matrix dictates. In this paper, we take a different route and apply an orthogonal DFT basis as it is used in OFDM, but subdivide its image into so-called sub-channels and let each sub-channel take only a fraction of the load. In a random fashion the subdivision is consecutively applied over a suitable number of time-slots. Within the time-slots the users will not change their sub-channel assignment and send in parallel the data. Activity detection is carried out jointly across time-slots in each of the sub-channels. For such system design we derive three rather fundamental results: i) First, we prove that the subdivision can be driven to the extent that the activity in each sub-channel is sparse by design. An effect that we call sparsity capture effect. ii) Second, we prove that effectively the system can sustain any overload situation relative to the DFT dimension, i.e. detection failure of active and non-active users can be kept below any desired threshold regardless of the number of users. The only price to pay is delay, i.e. the number of time-slots over which cross-detection is performed. We achieve this by jointly exploring the effect of measure concentration in time and frequency and careful system parameter scaling. iii) Third, we prove that parallel to activity detection active users can carry one symbol per pilot and time-slot so it supports so-called one-shot messaging. The key to proving these results are new concentration results for sequences of randomly sub-sampled DFTs detecting the sparse vectors “en bloc”. Eventually, we show by simulations that the system is scalable resulting in a coarsely 20-fold capacity increase compared to standard OFDM.