Almost Invariant Sets Under Affine Maps

Definitions and mathematical statement

For a positive integer $p$, let $G={\mathbb{Z}}/p{\mathbb{Z}}$. For a subset $A\subseteq G$, its density is $|A|/p$. The supplied hypothesis is exact density one half: \[ \frac{|A|}{p}=\frac12. \] For integer coefficients $a,b$, the associated map on residue classes is \[ \phi_{a,b}:G\longrightarrow G,\qquad x\longmapsto ax+b\pmod p. \] This notation describes maps induced by integer lifts; the record does not specify whether the bounds $|a|,|b|\leq K$ concern such lifts or a chosen absolute-value convention for residue classes.

Source question. Under what conditions on $K$ can a set $A$ of density $1/2$ be almost invariant under every map $\phi_{a,b}$ with $|a|,|b|\leq K$? The universal quantifier over the stated affine maps is retained, but "almost invariant" and the regime of $K$ are not defined in the supplied statement.

For clarity about the missing definition, the symmetric difference of sets $B,C$ is $B\triangle C=(B\setminus C)\cup(C\setminus B)$. Two possible quantities that one could consider are \[ \frac{|A\triangle\phi_{a,b}(A)|}{p} \quad\text{and}\quad \frac{|A\triangle\phi_{a,b}^{-1}(A)|}{p}. \] Here $\phi^{-1}(A)$ denotes a preimage, not necessarily application of an inverse map. The source selects neither of these errors, nor a normalization by $|A|$, nor an allowable error threshold. For bijections the two displayed errors agree, but the source has not imposed bijectivity: the displayed coefficient range includes $a=0$ whenever $K\geq0$. More generally, bijectivity requires $a$ to be invertible modulo $p$.

There is also a compatibility issue in the density hypothesis. The equality $|A|=p/2$ requires $p$ to be even. If $p$ is intended to be an odd prime, no set satisfies the literal hypothesis. The source does not explicitly state primality or replace exact half-density by an approximation, and this draft makes neither change.

A precise target would need to settle these issues and specify whether existence of suitable sets $A$, a statement about every half-density set, or a characterization of particular sets is sought; it must also fix the error scale, dependence on $p$, and desired range of $K$. The supplied literature triage dated 2026-08-17 itself flags the missing error metric and quantifier scale, reports that formal variants are addressed by approximate group-action and affine-invariance methods, and says no exact theorem matching the broad wording was found. Its partially solved label and the registry's open label are supplied classifications, not an independent theorem-status assessment.

Short English statement

Ask when a half-density subset of a cyclic group can change very little under every affine map with small coefficients. The error notion, size regime, and quantifiers on the set are missing. Exact half-density also conflicts with odd group order, and the coefficient range includes noninvertible constant maps.

Sources

• UnsolvedMath local registry, data/unsolvedmath/problems.json, permanent ID 144, source code catalogue entry, "Almost Invariant Sets Under Affine Maps," in Ben Green's 100 Open Problems collection.
• B. Green, 100 Open Problems, described in the supplied review as updated January 2026: https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf. The review checked on 2026-08-17 cites this collection for the agenda-level wording. No particular external problem number is supplied for this record, and the PDF was not accessed for this draft.
• Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems; local record including literature triage dated 2026-08-17. Dataset curation and original metadata are supplied under CC BY 4.0; referenced source material retains its own terms.