I'm still experimenting with this, so take that as a note of caution.

I lowered the alkalinity in our pool from 220ppm to 100ppm and still had the pH bounce back up to 7.6 each day after adding muriatic acid the day before to get it down to 7.2. So I lowered the alkalinity to 70-80ppm and now the pH is holding at 7.2 over the last two days. Be careful with the amount of acid added as you approach the lower alkalinity levels, a little has a bigger effect on the pH and it doesn't bounce back as far.

I'm not sure of the chemistry involved, but I suspect there's a CO2 interaction ocurring at the water surface/atmosphere interface that sets the pH dependent on the alkalinity.