Good information, @Isaac Zachary.
I'll add a few bits:
As you say, atmospheric air is effectively inert, though EGR has far less oxygen in it to combine with nitrogen, so it naturally produces less NOx. On the other hand, if you have less oxygen, you can have a harder time igniting a lean charge.
One of the major efficiency savings is reportedly in eliminating vacuum.
I'm most familiar with Honda's lean burn tactics. Their EGR systems tend to need manual cleaning every 100-150k miles. The G1 Insight runs around 25:1 AFR. Honda has a direct injection lean burn engine in the K20B, which is supposed to run as lean as 65:1 under certain circumstances (wikipedia, citation needed). Running much leaner than 19:1 or so needs combustion chamber tricks. E.g. The Insight uses a swirl tactic that passes a stream of fuel over a clocked spark plug, whereas the K20B based direct injection system ignites the fuel before it has a chance to spread out and get too diffuse to ignite - something you can't do without direct injection. Having an offset crankshaft helps in getting the crank angle optimal for the much slower burn rate of a lean burn engine, which can otherwise cause efficiency and power loss. Broadly speaking, a faster burn rate is more efficient, and lean burn slows it down, so it comes with downsides.
Atkinson cycle can also reduce vacuum, while also increasing the expansion ratio. Arguably it's better than lean burn, though you can probably use the two together.
Mazda published this helpful chart with regards to their experiments in Atkinson cycle and compression ignition:
Read further here:
https://www.insidemazda.co.uk/2017/1...-x-technology/