Efrat Bank‘s interesting number theory seminar here before break was about sums of arithmetic functions on short intervals in function fields. As I was saying when I blogged about Hast and Matei’s paper, a short interval in F_q[t] means: the set of monic degree-n polynomials P such that

deg(P-P_0) < h

for some monic degree-n P_0 and some small h. Bank sets this up even more generally, defining an interval in the space V of global sections of a line bundle on an arbitrary curve over F_q. In Bank’s case, by contrast with the number field case, an interval is an affine linear subspace of some ambient vector space of forms. This leads one to wonder: what’s special about *these specific* affine spaces? What about general spaces?

And then one wonders: well, what classical question over Z does this correspond to? So here it is: except I’m not sure this is a classical question, though it sort of seems like it must be.

**Question:** Let c > 1 be a constant. Let A be a set of integers with |A| = n and max(A) < c^n. Let S be the (multi)set of sums of subsets of A, so |S| = 2^n. What can we say about the number of primes in S? (**Update: **as Terry points out in comments, I need some kind of coprimality assumption; at the very least we should ask that there’s no prime factor common to everything in A.)

I’d like to say that S is kind of a “generalized interval” — if A is the first n powers of 2, it is *literally* an interval. One can also ask about other arithmetic functions: how big can the average of Mobius be over S, for instance? Note that the condition on max(S) is important: if you let S get as big as you want, you can make S have no primes or you can make S be half prime (thanks to Ben Green for pointing this out to me.) The condition on max(S) can be thought of as analogous to requiring that an interval containing N has size at least some fixed power of N, a good idea if you want to average arithmetic functions.

Anyway: is anything known about this? I can’t figure out how to search for it.