Weak inhomogeneous Duffin–Schaeffer approximation

Source and verification

This article explains family 022 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Approximation with a fixed shift

For $t\in \mathbb{R}$, write $\Vert t\Vert ={\min }_{a\in \mathbb{Z}}|t-a|$. Fix a real shift $\gamma$ and a finite-valued function $\psi :\mathbb{N}\to [0,\infty )$. The inequality

$$\Vert qx-\gamma \Vert <\psi (q)$$

means that $x$ lies within $\psi (q)/q$ of some point $(a+\gamma )/q$. Each denominator therefore gives a periodic array of approximation intervals.

Let $W_{\gamma }(\psi )$ be the set of points satisfying the inequality for infinitely many $q$. This is a limsup set: a point must belong to arbitrarily late unions of the approximation intervals.

Claimed weak inhomogeneous Duffin–Schaeffer theorem

For every fixed $\gamma$, if

$$\sum _{q\ge 1}\frac{\phi (q)}{q}\psi (q)=\infty ,$$

then $W_{\gamma }(\psi )$ has full Lebesgue measure. No monotonicity of $\psi$ is assumed. The exceptional null set may depend on both $\gamma$ and $\psi$. (OpenAI, 2026)

Here $\phi (q)$ counts integers coprime to $q$. The numerator $a$ in the approximation is unrestricted. The word “weak” refers to that numerator convention; this is not a coprime-numerator theorem.

🏷️ Divergence and dependence

For $\psi (q)\le 1/2$, the approximation set in $[0,1)$ has measure $2\psi (q)$, since its $q$ intervals have disjoint interiors and each has length $2\psi (q)/q$.

If ${\sum }_q\psi (q)<\infty$, the first Borel–Cantelli lemma gives a null limsup set. Divergence is harder: events with infinite total mass can still overlap so much that their limsup has small measure.

The totient-weighted condition is stronger than unweighted divergence and compensates for arithmetic overlap among denominator grids. Since $\phi (q)/q\le 1$, it implies ${\sum }_q\psi (q)=\infty$, but the theorem does not claim the converse suffices.

It also gives no convergence converse when only the weighted sum converges. The distinction is important because unrestricted numerator grids have total measure governed by $\psi (q)$, not by $\phi (q)\psi (q)/q$.

🏷️ A second-moment mechanism

For a nonnegative random variable $Z$, Cauchy–Schwarz gives

$$\mathbb{P}(Z>0)\ge \frac{(\mathbb{E}Z{)}^2}{\mathbb{E}{Z}^2}.$$

Applied to a weighted sum of interval indicators, this turns mass and overlap estimates into a positive-measure union. Full measure requires a localized version or a separate argument excluding every positive-measure avoided set.

The source works with arbitrarily late finite blocks whose totient-weighted masses lie in a fixed small interval. It constructs a retained nonnegative sum inside the genuine approximation intervals and a virtual comparison sum.

The comparison is equidistributed on fixed spatial intervals; the expected discarded mass is small; and the retained sum has bounded second moment. If a positive-measure set avoided all sufficiently late rows, approximate it by one fixed finite union of intervals. Equidistribution controls that finite union, and the second moment controls the approximation error. This yields the full-measure conclusion.

Fixing the approximating union before taking the late-block limit avoids treating an arbitrarily changing measurable set as a uniform-distribution test.

🏷️ Why the overlap estimate is arithmetic

Two nearby grid points satisfy

$$\left|\frac{a+\gamma }{q}-\frac{b+\gamma }{r}\right|\text{small}.$$

After multiplication by $qr$, this constrains $ra-qb+(r-q)\gamma$. For irrational $\gamma$, this is not simply an integer congruence.

The first arithmetic obstruction is concentration of denominator prime exponents near one common integer. An inverse estimate extracts such concentration when a weighted least-common-multiple estimate fails. Varying permitted factors then either disperses the projected phases or forces useful rational approximations to them. The construction imposes numerator restrictions adapted to these rational models, without imposing global coprimality on the final theorem.

The second obstruction is recurring alignment through large common primes. The source exposes prime factors in stages, keeping a projected grid finer than the eventual true row. Linked portions of weights are suppressed when a divisibility test succeeds, preserving first-moment normalization while canceling dangerous transfers.

Labels retain their previous connections even when intermediate labels disappear. A squared-mass budget bounds the total covariance at the first interaction of previously separate collections. These averages use common residue translations; they do not assume that divisibility tests are independent or randomize the prescribed shift.

🏷️ Hausdorff-measure consequence

For a continuous nondecreasing dimension function $f$ with $f(r)\to 0$ and monotone $f(r)/r$, the source applies mass transference. If

$$\sum _{q\ge 1}\phi (q)f\left(\frac{\psi (q)}{q}\right)=\infty ,$$

then every bounded interval $I$ satisfies ${\mathcal{H}}^f(I\cap W_{\gamma }(\psi ))={\mathcal{H}}^f(I)$. This is again a sufficient divergence criterion with unrestricted numerators. The source does not assert a quantitative count of solutions.

🏷️ Key steps and proof outlines

Reduce divergence to arbitrarily late blocks of controlled mass

Write $r(q)=\psi (q)/q$ for the approximation radius in the $x$ variable. First consider the easier case in which $qr(q)=\psi (q)$ does not tend to zero.

Proof outline. Along an unbounded sequence, $r(q)\ge c/q$ for some fixed $0<c<1/2$. The union of radius-$c/q$ intervals around the shifted grid has mean $2c$. On a fixed interval $J$, its integral is $2c|J|+O(1/q)$, because only two incomplete grid periods cause error. Approximate any fixed measurable set by a finite union of intervals; boundedness of the indicators transfers this convergence to that set. Hence no positive-measure set can avoid the entire tail. Taking the countable union of all tail-avoiding sets gives the full-measure conclusion.

Otherwise $qr(q)\to 0$. Round positive radii down to dyadic values between half their original value and their original value. Divergence survives. For a fixed small $w>0$, begin arbitrarily far out and stop a block when ${\sum }_q\phi (q)r(q)$ first reaches $w$. Each late summand is at most $w$, so the block mass lies in $[w,2w]$. This supplies finite objects of comparable mass even though no monotonicity of $\psi$ is available. (OpenAI, 2026)

A prime update preserves mass under one common translation

The construction carries weighted pieces of projected grids through successive prime updates. A cap $N_0$ limits the total raw load in each cell. Plus and minus lists are matched before the update; the same core translation moves the entire configuration.

Proof outline. For a matched portion of size $m$, after including the prescribed arithmetic scaling, the two masses before the update are $m/p$ and $m(1-1/p)$. Afterwards they are $mS$ and $m(1-S)$, where $S$ is the divisibility-success indicator. Translation by one step cycles a numerator through the $p$ residues, so the orbit average of $S$ is exactly $1/p$. Thus each transported mass has the correct marginal mean. Matched pieces contribute their full combined mass, while unmatched pieces occur only on the larger side and can only decrease. The cell load after updating is therefore at most the larger load before updating, preserving the cap. This proves a first-moment invariant without treating different success indicators as independent Bernoulli variables.

A squared-load budget pays for repeated interactions

Labels remember which pieces have already become connected. Even if a particular label later disappears, its inherited equivalence information is retained. The source’s accounting energy has the form ${\sum }_A{r}_A\lambda (A{)}^2$, with $\lambda (A)$ the load of a retained class and $r_A$ its prescribed weight.

Proof outline. Averaging an update over its common translation gives a variance increment of the form ${\sum }_A{r}_A\mathrm{Var}(Y_A)\ge 0$. Merging classes adds nonnegative cross terms to squared loads. The cap bounds squared mass by $N_0$ times first mass, so the accumulated increments are bounded by $O(N_{0}W)$, with $W$ the block mass; discarded mass is charged by the source’s telescoping identity. Keeping inherited equivalence is what prevents an artificial split from producing a negative energy change and making an old interaction appear new. Combined with the arithmetic overlap estimates, this budget bounds the final broad overlap energy and hence the $L^2$ norm of the realized nonnegative sum. The arithmetic estimates and partition construction are major inputs, not consequences of the elementary variance identity alone.

The finite-block contradiction gives full measure directly

Suppose a measurable set $E$ of measure $\mu >0$ avoids every sufficiently late original row. Let $F^{\mathrm{vir}}$ be the virtual comparison sum and $F$ the retained sum, whose support lies inside the genuine approximation intervals. The source proves, with constants fixed before choosing the late block,

$$\begin{array}{c}{M}_{-}\le V:=\mathbb{E}\int F^{\mathrm{vir}}\le M_{+},\\ \mathbb{E}\Vert F^{\mathrm{vir}}-F{\Vert }_1\le {\eta }_{s}V+\mu M_{-}/20,{\eta }_s\le \mu /20,\\ \Vert \mathbb{E}F{\Vert }_2\le C_2.\end{array}$$

Proof outline. Choose a fixed finite union of intervals $U$ approximating $E$, with

$$|U△E|\le \min \left(\mu /2,(\mu M_{-}/(4C_2){)}^2\right).$$

For sufficiently late blocks, equidistribution gives $\mathbb{E}{\int }_U{F}^{\mathrm{vir}}\ge 0.4\mu V$. Subtracting the loss bound yields ${\int }_U\mathbb{E}F\ge 0.3\mu M_{-}$. But $F$ vanishes on $E$, so Cauchy–Schwarz gives

$${\int }_U\mathbb{E}F\le C_2|U△E{|}^{1/2}\le 0.25\mu M_{-}.$$

The contradiction rules out every positive-measure tail-avoiding set. Two details make the deduction valid. The soft loss is proportional to the actual virtual mass $V$, so one does not need to choose its parameter after discovering a possibly small lower bound. And $U$ is selected after $C_2,M_{-}$ are fixed but before the late-block limit, so equidistribution is used only on a fixed test set.

🏷️ Connections

Approximation to pi concerns one specified number, while the present theorem is metric, holding for almost every $x$. Jacobsthal covering offers a finite congruence-survival problem in which worst-case dependence is also central.

🏷️ Manuscripts and verification scope

• The Weak Inhomogeneous Duffin—Schaeffer Conjecture — pinned manuscript and companion source.

No family scope page was present at lean/docs/022.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. The Weak Inhomogeneous Duffin–Schaeffer Conjecture.