Research log — thin cyclic additive bases

Status (accessed 2026-08-17)

Green, 100 Open Problems, Problem 33 (Dec 2025 update): OPEN. “Potentially yes, but I’m not sure I know how to construct such sets.” Prime-modulus variant equivalent to a near-optimal direction set on the parabola in $\mathbb F_p^2$ .
Granville via Croot–Lev, Problem 5.2 (CRM 43, 2007): smallest direction-determining set in $\mathbb F^r$ . Counting $\sqrt2\,q^{(r-1)/2}$ ; Konyagin matches up to a constant via a difference basis.
Caprace–de la Harpe, arXiv:1807.04992, Confluentes Math. 12 (2020): $n_p>\sqrt{2p}$ , $n_p\le2\lceil\sqrt p\rceil+1$ (Fitch–Jamison). Exact $n_p$ open.

Published record on $\mathrm{SS}(n,2)=\min\{|A|:A+A=\mathbb Z/n\mathbb Z\}$

quantity	bound	source
all $n$ , counting	$\mathrm{SS}(n,2)\ge(-1+\sqrt{1+8n})/2$	pair count
all $n$ , elementary	$\mathrm{SS}(n,2)<2\sqrt n+O(1)$	interval + AP
all large $n$	any $c>\sqrt3\approx1.732$	Jia–Shen, SIDMA 31 (2017)
infinitely many $n$	$\liminf\mathrm{SS}(n,2)/\sqrt n\le\sqrt{8/3}\approx1.63299$	Bevan–Erskine–Lewis, arXiv:1506.04962, Cor. 18
interval bases (hence cyclic)	$\sqrt{3.5}\approx1.871$	Mrose
difference cover	$\Delta[\mathbb Z_n]/\sqrt n\to1$ along Singer/Bose	Banakh–Gavrylkiv arXiv:1702.02631
exact small $n$	cyclic through $n=67$ at size 13; all abelian through 85	Haanpää, JIS 7 (2004); Fitch–Jamison through 54

No later improvement of BEL’s liminf, or of Jia–Shen’s universal $\sqrt3+\varepsilon$ , found after those papers (search through 2026-08-17, including Lewis’s 2021 thesis and Erskine’s later degree-diameter work). Vetrík’s $L^-(2)\ge8/9$ is for general directed Cayley graphs, not cyclic groups; it would violate the sum-cover counting bound if it were cyclic.

Independent checks (this folder)

python3 compute/make_bel_certs.py
python3 compute/haanpaa_replay.py
python3 compute/verify.py

BEL $q=7,13,19,31,37,43,61$ : product-group cover and cyclic A+A both hold. Ratios $1.932\to1.660$ , consistent with $\sqrt{8/3}$ . Certificates in compute/certs/bel_q*.json. Greedy deletion on $q=13,19$ drops nothing.
Haanpää Table 3, cyclic rows: all twelve covers verify. Isolated small- $n$ ratios $1.15$ – $1.59$ are not a dent.

What we did not beat

Green’s $(\sqrt2+o(1))\sqrt q$ for infinitely many $q$ .
BEL’s liminf $\sqrt{8/3}$ .
Jia–Shen’s universal $\sqrt3+\varepsilon$ .
Caprace–de la Harpe’s exact $n_p$ .

Failed templates (logs in compute/): equal-length 3-AP families; Singer/Bose interval and AP repairs; Singer greedy completion; two-AP deletion; quadratic windows; geometric progressions; local search at Haanpää scale. Search residue is not a bound.

Did not claim

The Green problem. An improvement of $\sqrt{8/3}$ . A finite table as an asymptotic density.