[Ethereum] How do we use block.number (as opposed to block.timestamp) for measuring time

I believe the block.timestamp is the wall-clock date/time at which the block was mined. A block number set in the future based on some other block is based on an estimate of the rate of block mining. For example, the average/target block time is, at the time of writing, about ~17s. So if you want to set a time 24 hours in the future via block number, you can use 24 * 60 * 60 / 17 (number of seconds in a day divided by number of seconds per block) to determine roughly what block number will be mined at that time. Note that for times in the mid- and far- future, you will need to take into account the ice age: the rate of block generation will slow down, by design, to encourage a hard fork for proof-of-stake and, in the meantime, makes the profitability of mining decrease materially (when denominated in eth) as the ice age progresses.

Here is a simulator to get an idea of what is happening: https://gist.github.com/CJentzsch/c78768f9837afb8eef74 Note that the rate of slowing has been slowed by a more recent change to the Ethereum protocol, so the simulator is out of date.

You use signing in the same way that you would if one person was signing. Each person will have to pay the transactions costs needed to submit their signature. However you could create a single function that will automatically parse all signed messages for slightly cheaper than if you were to submit a new transaction to parse each signed message. You could potentially implement something like merkle airdrops (https://blog.ricmoo.com/merkle-air-drops-e6406945584d)

There are a few gotchas that come with verifying signed messages, I have a set of contracts and python signer program which demonstrate the workflow needed to sign messages:

When signing messages they get prepended with the following data: https://github.com/postables/Postables-Payment-Channel/blob/7d2f91bb060f80b139cab72b5fdff79d116f6210/solidity/ChannelsV4.sol#L12

I also have a python script which can be used to sign messages: https://github.com/postables/Postables-Payment-Channel/blob/develop/python/signer.py

The following solidity function details how to verify the ORIGINAL message that was signed, along with the signer of the signed message: https://github.com/postables/Postables-Payment-Channel/blob/7d2f91bb060f80b139cab72b5fdff79d116f6210/solidity/ChannelsV4.sol#L257

The following solidity function details how to verify the signer of a message: https://github.com/postables/Postables-Payment-Channel/blob/7d2f91bb060f80b139cab72b5fdff79d116f6210/solidity/ChannelsV4.sol#L192