I looked around, and did a bit of experimentation, and discovered the following.
What gets saved automatically when you exit
The game saves information about which Zone and Act you've reached. It also saves information about how many Chaos Emeralds you've collected, and how many lives you have left.
The game does not save your progress within an Act. Even if you've passed a "Star Post" checkpoint, it doesn't matter. When you reload your save game, you'll go back to the beginning of the Act.
How to avoid losing your progress within an Act
If you don't want to lose your progress within an Act:
You can pause the game, and can go back to the Android home screen, and can leave the game paused for hours or even overnight. If you use too much RAM in one or more other apps, the game may be kicked out of RAM, and you may have to return to the beginning of the Act. But, if not, the game will stay in RAM, and you can resume exactly where you left off.
If you have a flagship Samsung phone released in the past five years or so, you can lock the game in RAM, so that it will stay open indefinitely. However if you use this ability, the game might take up perhaps 300 to 600 MB of your phone's RAM, all day, every day, even if you haven't played it for months or years.
If the game crashes or is kicked out of RAM
If the game crashes, or if it's kicked out of RAM while you use other apps: All new Chaos Emeralds gained, and all other progress made since the last time the game was saved, will be lost.