Arithmetic removal in cyclic groups: corrected quantifiers, matching distances, and exact CRT-compatible lifts

Definitions and mathematical statement

For $m\geq1$, let $G_m=\mathbb Z/m\mathbb Z$, written additively. For three subsets $A,B,C\subseteq G_m$, define the ordered solution count \[ T(A,B,C)=|\{(a,b,c)\in A\times B\times C:a+b+c=0\}|. \] The three sets are distinguished roles: deleting a residue from one does not automatically delete it from another. For $0<\varepsilon<1$, let $\delta_{\rm cyc}(\varepsilon)$ be the supremum of those $\delta\in[0,1]$ for which the following assertion holds for every $m$ and every such triple. If $T(A,B,C)\leq\delta m^2$, there exist $A'\subseteq A$, $B'\subseteq B$, $C'\subseteq C$ such that \[ |A\setminus A'|,\ |B\setminus B'|,\ |C\setminus C'| \leq\varepsilon m, \qquad T(A',B',C')=0. \] Cardinality inequalities have their literal integer meaning, so no rounding convention is needed. The problem is to determine, or give substantially sharper quantitative estimates for, $\delta_{\rm cyc}(\varepsilon)$ as $\varepsilon\downarrow0$, uniformly over all cyclic group orders. Upper bounds require examples far from solution-free despite few solutions; lower bounds must give a deletion guarantee for every triple. The source's "at least" solution hypothesis is reversed here to "at most", as required by a removal lemma. Its final reference to regularity is interpreted in the context of the explicitly stated arithmetic removal problem.

Short English statement

Quantify how few additive triples force three subsets of a cyclic group to become solution-free after small deletions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000895 (AIM-COMBINATORICS-0020), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Additive combinatorics and its applications, item 1.4. Original workshop question underlying AIM-COMBINATORICS-0020. http://aimpl.org/addcombapp/1/.