Asymptotics of sum-representation extrema over prime fields

Determine the asymptotic behavior of \(\beta_2(\mathbb{F}_p)\) and \(\nu_2(\mathbb{F}_p)\) even along sequences of prime values of \(p\).

Background

The paper turns from difference bases to additive bases and bounded sum representations. For the one-dimensional vector space Fp\mathbb{F}_p, the authors state that even the asymptotics of the extremal bounded-representation quantity β2(Fp)\beta_2(\mathbb{F}_p) and the minimum additive-basis quantity ν2(Fp)\nu_2(\mathbb{F}_p) are not known along any sequence of primes. These problems are presented as part of the broader unresolved theory of sumsets over finite fields.

References

When n = 1, it is unknown for even a sequence of values of p the asymptotics of β2(Fp) or ν2(Fp) and determining these are listed in [18] in the discussion after Problem 31 and in Problem respectively.

— Cardinalities of $g$-difference sets  (2501.11736 - Schmutz et al., 20 Jan 2025) in Section 5.1, page 15