The emerging blockchain protocols provide a decentralized architecture that is suitable of supporting Internet of Things (IoT) interactions. However, keeping a local copy of the blockchain ledger is infeasible for low-power and memory-constrained devices. For this reason, they are equipped with lightweight software implementations that only download the useful data structures, e.g. state of accounts, from the blockchain network, when they are updated. In this paper, we consider and analyze a novel scheme, implemented by the nodes of the blockchain network, which aggregates the blockchain data in periodic updates and further reduces the communication cost of the connected IoT devices. We show that the aggregation period should be selected based on the channel quality, the offered rate, and the statistics of updates of the useful data structures. The results, obtained for the Ethereum protocol, illustrate the benefits of the aggregation scheme in terms of a reduced duty cycle of the device, particularly for low signal-to-noise ratios, and the overall reduction of the amount of information transmitted in downlink (e.g., from the wireless base station to the IoT device). A potential application of the proposed scheme is to let the IoT device request more information than actually needed, hence increasing its privacy, while keeping the communication cost constant. In conclusion, our work is the first to provide rigorous guidelines for the design of lightweight blockchain protocols with wireless connectivity.