Progression lifting gives a four-dimensional shrinking-gap family

Definitions and mathematical statement

This record is historical context rather than the statement of Conjecture 34. The aid identifies its first sentences with a preceding positive-definite shrinking-gap question, and its following sentences with the distinct indefinite Oppenheim theorem. The source's "indefinite definite" wording is contradictory and is not adopted as a mathematical hypothesis.

The supplied special-case theorem is precise. Let $P$ be an integer-valued positive-definite quadratic form on ${\mathbb{Z}}^r$. Suppose there are integers $M\geq1$, a residue $a\pmod M$, and $N_0$ such that every integer $N\geq N_0$ with $N\equiv a\pmod M$ is represented by $P$. For every irrational $\alpha>0$, set \[ Q(y,n)=P(y)+\alpha n^2,\qquad(y,n)\in{\mathbb{Z}}^r\times{\mathbb{Z}}. \] Since $Q$ is positive definite, its distinct integral values form a locally finite increasing sequence $v_0<v_1<\cdots$ tending to infinity. The aid asserts \[ v_{j+1}-v_j\longrightarrow0. \] This concerns consecutive distinct values, not multiplicities. Equivalently, sufficiently far out, every interval of any fixed positive length meets the value set. No Diophantine condition on $\alpha$ beyond positivity and irrationality is imposed.

For the supplied four-dimensional family, take $P(y)=y_1^2+y_2^2+y_3^2$. Legendre's three-square theorem represents every positive integer congruent to $1\pmod8$, so the progression hypothesis holds and \[ Q(x)=x_1^2+x_2^2+x_3^2+\alpha x_4^2 \] has shrinking gaps for every irrational $\alpha>0$. This does not establish the corresponding assertion for every positive-definite quaternary form.

The original source's historical status claim is that the preceding Conjecture 34 was proved in dimensions $d\geq9$ in [15], then extended by that method to $d\geq5$, while $d=3,4$ were still open. The full preceding conjecture is not reproduced, and these are source-era assessments, not an independent current-status claim.

Separately, the Oppenheim discussion is read in its intended indefinite sense: if $Q$ is a nondegenerate indefinite real quadratic form in dimension $d\geq3$ and is not a nonzero real multiple of a rational form, the source attributes to Margulis [129] the density \[ \{Q(x):x\in{\mathbb{Z}}^d\}\text{ is dense in }{\mathbb{R}}. \] It says that proof was not effective. That indefinite density theorem is not the positive-definite shrinking-gap assertion and is not presented as a solution of the missing Conjecture 34. The supplied progression result is only a special family within that latter program.

Short English statement

The source mixes historical shrinking-gap status with a separate indefinite theorem. The supplied progression criterion proves shrinking gaps for one explicit four-dimensional family for every positive irrational coefficient. It does not solve all quaternary forms, and indefinite density is kept separate.

Sources

• Local UnsolvedMath registry, permanent ID $20003320$, exact source code catalogue entry; canonical aim-other-notes.json, notes[126], AIM domain Other, registry category Miscellaneous, source tag conjecture.
• AIM workshop "Emerging applications of measure rigidity," assigned item 34 context: https://aimath.org/WWN/measrigid/measrigid.pdf. Statement status reconstructed_unverified; the record contains historical prose rather than the preceding conjecture. No PDF was fetched.
• Source-local [15] concerns the historical positive-definite dimensional result; [129] is attributed to Margulis for Oppenheim. The aid invokes Legendre's three-square theorem for its special case. Full authoritative bibliographic entries are not supplied. The original literature field is empty.
• Research attempt 1: result context_only, problem status at run not_a_problem, rigor label proved, novelty confidence low; classification OPEN-TRIAGE, registry status open, suggested difficulty L2. Created and updated 14 August 2026. These are supplied assessments, not a resolution of every dimension-four form.
• Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. The machine-generated research aid is not peer reviewed and requires independent verification before citation.