Structural Thresholds for Sum-Free Sets in $\mathbb{Z}/p\mathbb{Z}$

Definitions and mathematical statement

For a prime $p$, a set $A\subset{\mathbb{Z}}/p{\mathbb{Z}}$ is sum-free if $(A+A)\cap A=\varnothing$, allowing the two summands to coincide. Its density is $|A|/p$. For an integer $1\le n\le p/2$, let \[ I_{p,n}=\{n,n+1,\ldots,p-n\}\pmod p. \] A dilation is multiplication by $u\in({\mathbb{Z}}/p{\mathbb{Z}})^\times$; no translation is included.

The source asks for the sharp structural threshold, between the historical cardinality scales $0.2p$ and $0.33p$, above which every sum-free set is contained in some $uI_{p,n}$. The dimensionless densities are $0.2$ and $0.33$; the factors $p$ refer to set sizes. The decimal $0.33$ is retained and not silently replaced by $1/3$.

The extract does not define $n$. A natural intended strong version is $n=|A|$, leading to the property \[ |A|>\gamma p\quad\Longrightarrow\quad \exists u\in({\mathbb{Z}}/p{\mathbb{Z}})^\times,\ A\subset uI_{p,|A|} \] for all sufficiently large primes and all sum-free $A$. In that asymptotic interpretation, one seeks the infimum of constants $\gamma$ for which this property holds uniformly. This is an explicit interpretation, not a verified recovery of the omitted definition of $n$ or of strict versus weak threshold endpoints. If $n$ were freely chosen, $n=1$ would make containment essentially trivial since a nonempty sum-free set excludes zero; thus its relation to $A$ is essential. The source also does not settle exact finite-prime versus asymptotic quantifiers.

The literature assessment checked on 21 August 2026 says the original $0.33$ threshold was improved in later work and mentions lower examples near $0.2$, while the full structural threshold remains unresolved in that assessment. No improved numerical constant or exact lower bound is supplied. The question concerns the stated interval-containment structure, not merely the maximal possible size of a sum-free set. These are dated triage claims requiring independent verification.

Short English statement

Find the sharp density forcing a sum-free set modulo a prime into the specified dilated central interval. The interval parameter must be tied to the set, presumably its size; otherwise the containment condition loses its intended structural force.

Sources

• Finite Fields: Theory and Applications (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/54, pp. 2913--2970, https://doi.org/10.4171/owr/2004/54; proposed year supplied as 2004.
• Literature assessment checked 21 August 2026: original OWR report, https://doi.org/10.4171/owr/2004/54; Candela--Gonzalez-Sanchez--Grynkiewicz, https://doi.org/10.24033/bsmf.2827.
• Local UnsolvedMath registry, permanent ID $30000154$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. Corrected-and-verified status records balanced group, density and interval notation; it does not supply the missing relation defining $n$.