I have a new Honda Fit Sport AT. 2,860 miles.
I am not getting the advertised mileage for city driving.
One person at my local Honda says mileage improves with enging wear-in.
Another person (service kid) said Nope... nothing will change.
I'm driving many different attempts to maximize variables. (low rpm, coasting, etc).
My question: Do engines improve fuel economy after the first few thousand miles or so?

Thanks!

Not in my experience. If you want to increase your gas mileage 8 to 10%, use synthetic engine oil and transmission fluid.

Quick answer is yes. The magic number is around 5,000mi. Also, this time of year is a bad time to get max milage figures. Fuel qualities improve and engines run more efficiently in warmer weather. And just last night, news reports said EPA figures that were put on cars were too high, and now all will be lowered to corespond with today's driving styles.
Give it some time. MPG will improve. Best wishes.