Exponential upper and lower bounds for the additive basis order of multiplicative subgroups

Definitions and mathematical statement

For a prime $p$, let $\mathbb F_p^*$ denote its multiplicative group. A multiplicative subgroup $H\leq\mathbb F_p^*$ is nonempty and closed under multiplication and inverses. For an integer $k\geq1$, its additive $k$-fold sumset is \[ kH=\{h_1+\cdots+h_k:h_i\in H\}\subseteq\mathbb F_p. \] Repetitions are allowed, but zero is not available as a summand unless it is represented by several elements of $H$. Thus $kH$ means exactly $k$ summands, rather than at most $k$.

Fix $\delta\in(0,1)$. Determine the smallest integer $K(\delta)$ for which \[ \forall p\text{ prime}\ \forall H\leq\mathbb F_p^*: |H|>p^\delta\ \Longrightarrow\ K(\delta)H=\mathbb F_p, \] allowing the value $\infty$ if there is no such integer. In particular, seek sharp dependence on $\delta$ as $\delta\downarrow0$ and matching examples requiring many summands. The parameter must be uniform in both the prime and the choice of subgroup. Once $kH$ equals the entire field, every larger number of summands also gives the entire field, since $H$ is nonempty. Consequently the least uniform order is a coherent threshold despite the exact-summand convention. The source gives an existence bound in terms of $\delta$ and asks for improvement; the target here is the optimal additive-basis order rather than a claim that the historical estimate is best known.

Short English statement

How many additive summands from a multiplicative subgroup of size greater than a fixed power of the field size are uniformly sufficient to cover the prime field?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001114 (AIM-COMBINATORICS-0239), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Recent trends in additive combinatorics, item 3.2. Original workshop question underlying AIM-COMBINATORICS-0239. https://aimath.org/WWN/additivecomb/additivecomb.pdf.