Rational approximation with any fixed shift

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Family 022 · Number theory
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

Suppose we try to approximate a real number $x$ by fractions, but move each numerator by the same fixed real shift $\gamma$. For denominator $q$, the available centers are

$$\frac{a+\gamma}{q},\qquad a\in\mathbb Z.$$

We may choose a different error tolerance for every denominator. Some denominators can receive a generous tolerance, others a tiny one, and most can be ignored altogether. Under what condition does almost every real number lie near infinitely many of these shifted fractions?

Family 022 claims the weak inhomogeneous Duffin–Schaeffer conjecture for every prescribed real shift. Its divergence condition uses Euler’s totient function to account for arithmetic redundancy among denominators. It requires no monotonicity or regular spacing of the tolerances.

The qualification “weak” matters: the integer $a$ is unrestricted. The theorem does not require $a$ and $q$ to be coprime.

2. Background and essential definitions

For a real number $t$, let

$$\|t\|=\min_{a\in\mathbb Z}|t-a|.$$

Then $\|qx-\gamma\|<\psi(q)$ means that some integer $a$ satisfies

$$\left|x-\frac{a+\gamma}{q}\right|<\frac{\psi(q)}q.$$

The nonnegative function $\psi:\mathbb N\to[0,\infty)$ supplies the tolerances. Finite-valued means each $\psi(q)$ is finite; the function need not be bounded globally. A zero value simply makes that denominator unavailable.

Euler’s totient $\varphi(q)$ counts integers between one and $q$ that are coprime to $q$. Its ratio

$$\frac{\varphi(q)}q=\prod_{p\mid q}\left(1-\frac1p\right)$$

measures the proportion of reduced residue classes. For $q=12$, these are $1,5,7,11$, so the ratio is $1/3$. Denominators divisible by many different small primes have a smaller ratio.

For each $q$, the approximation event is a periodic union of intervals centered at $(a+\gamma)/q$, each of radius $\psi(q)/q$. When $\psi(q)\le1/2$, its total length in one period is $2\psi(q)$. However, events for different denominators can overlap strongly. Summing their lengths does not by itself prove that infinitely many occur.

Lebesgue-almost every means all real numbers except a set of Lebesgue measure zero. It is a statement about length, not every individual real number. The exceptional set is allowed to depend on both $\gamma$ and $\psi$.

The relevant infinite-occurrence set is

$$W_\gamma(\psi)=\{x:\|qx-\gamma\|<\psi(q)\text{ for infinitely many }q\}.$$

It is a limsup set: a point belongs if it survives in approximation events arbitrarily far along the denominator sequence.

3. Development history and previous partial results

Khintchine’s theorem gives a clean answer in the homogeneous case $\gamma=0$ when $\psi$ is non-increasing: divergence of $\sum_q\psi(q)$ gives full measure. Szüsz extended the monotone theory to fixed inhomogeneous shifts. Removing monotonicity makes overlapping denominators a serious obstacle.

Duffin and Schaeffer exhibited the failure of a naive unweighted criterion in the homogeneous setting and proposed a theorem for reduced fractions, weighted by $\varphi(q)/q$. Koukoulopoulos and Maynard proved that conjecture in their 2020 Annals paper, cited by the manuscript. It is a major antecedent, but does not automatically settle arbitrary shifted centers.

Inhomogeneous overlap is difficult even when numerators are unrestricted. Ramírez constructed nonmonotone counterexamples to unweighted divergence for prescribed countable collections of shifts. These have finite totient-weighted sum, so they do not contradict the statement considered here. His primary paper explains the distinction.

Yu obtained results with additional divergence or size assumptions and restrictions on the shift. Chow and Technau proved the weak conjecture for certain structured approximation functions arising in multiplicative approximation. Beresnevich, Hauke and Velani proved the weak statement for every rational shift and obtained further irrational-shift results under additional conditions. Their 2024 paper is the immediate predecessor for the arbitrary-shift question.

There is also a crucial warning about the stronger numerator condition. The manuscript cites September 2026 counterexamples by Hauke-Treuer–Maynard–Pollington and independently He–Liao to the ordinary coprime-numerator formulation for broad classes of shifts. Thus “weak” is not a cosmetic adjective.

For a rational shift $\gamma=A/B$, the earlier positive theorem can impose the adapted condition $\gcd(A+aB,q)=1$. This is different from $\gcd(a,q)=1$. Family 022’s stated advance is to allow a completely arbitrary fixed irrational shift together with an arbitrary sparse, nonmonotone tolerance function, while leaving numerators unrestricted.

4. What the new manuscripts claim

The family contains one manuscript. Theorem 1.1 states that, for every fixed $\gamma\in\mathbb R$ and finite-valued $\psi$,

$$\sum_{q=1}^{\infty}\frac{\varphi(q)}q\psi(q)=\infty$$

implies

$$\|qx-\gamma\|<\psi(q) \quad\text{for infinitely many }q$$

for Lebesgue-almost every $x\in\mathbb R$.

There is no Diophantine restriction on $\gamma$. It may be rational, badly approximable, or extremely well approximated by rationals. No monotonicity is required of $\psi$, even on its support. The conclusion is a divergence implication, not an asymptotic count of solutions.

The quantifiers are worth keeping explicit: first fix $\gamma$ and $\psi$; then almost every $x$ works. The theorem does not give one universal exceptional null set valid simultaneously for all real shifts and all tolerance functions.

A Hausdorff-measure consequence

Corollary 1.2 extends the divergence statement to a finer measure of size. A gauge $f(r)$ assigns a cost to a covering interval of small diameter $r$; Hausdorff $f$-measure is obtained by minimizing total cost over increasingly fine covers. The familiar Hausdorff dimension uses gauges $f(r)=r^s$.

Assume $f$ is continuous, positive and nondecreasing on positive arguments, tends to zero as $r\downarrow0$, and has monotone ratio $f(r)/r$. If

$$\sum_{q=1}^{\infty}\varphi(q) f\!\left(\frac{\psi(q)}q\right)=\infty,$$

then for every bounded interval $I$,

$$\mathcal H^f(I\cap W_\gamma(\psi))=\mathcal H^f(I).$$

The corollary uses the mass transference principle of Beresnevich and Velani. It retains the fixed shift and unrestricted numerators, and asserts no convergence converse.

5. Why the advance matters

Provisional importance: 4/5. Taken as stated, this settles a major general form of metric approximation with an additive shift. Both sources of irregularity remain unrestricted: the shift can have any rational-approximation behavior, and the denominator support can be arbitrarily sparse.

The conclusion shows that the totient-weighted divergence condition is robust enough to overcome the resulting overlaps for unrestricted numerators. The Hausdorff consequence connects the theorem to fractal-size questions as well as ordinary length.

The mathematical significance lies in the uniformity of the hypotheses, not in a pointwise claim for every real number. This is a broad theorem about almost-everywhere approximation, with arithmetic correlations replacing the independence that elementary probability would prefer.

6. The main idea

Keep enough mass while controlling overlap

For independent events, divergent total probability often implies infinitely many occurrences. These approximation events are far from independent. A standard replacement is a second-moment estimate: if a nonnegative counting function has substantial average and its squared average is controlled, it cannot concentrate all its mass on an arbitrarily tiny set.

The paper works with arbitrarily late finite blocks of denominators whose totient-weighted mass lies in a fixed small range. Within each block it constructs a weighted sum of interval indicators, supported inside the original approximation events. It aims to retain enough first-moment mass while bounding the second moment.

A comparison sum follows the same prime-step updates but omits certain additional deletions. This sum is equidistributed on fixed spatial intervals. Small loss of mass and control of the retained second moment then rule out a positive-measure set avoiding every sufficiently late block. The argument approximates such an avoided set by one fixed finite union of intervals, rather than assuming an unproved zero-one principle for the shifted problem.

Close shifted fractions reveal denominator structure

Two centers can be close only when an integer combination of their numerators and shifted denominators is small. At very small scales there is too little room for direct averaging.

An inverse estimate shows that large weighted overlap forces the prime-exponent patterns of the denominators to cluster around a common integer. This builds on the common-pivot ideas from homogeneous Duffin–Schaeffer proofs, but the shifted phases require additional analysis.

Varying permitted factors then gives either a rotation estimate controlling the overlap or a precise rational model for the phase offsets. Exact coincidences in those models are handled by explicit numerator restrictions. Those restrictions are tools used on selected weighted subevents; the final theorem still permits all numerators.

Expose primes without pretending the tests are independent

Large common primes can create repeated correlations when previously separate collections of rows begin to interact. The manuscript assigns each denominator an integer multiple and exposes its prime factors in stages. Before all factors are exposed, a row is represented on a finer projected grid containing its eventual true centers.

At a prime step, nearby grid entries are grouped. Some weights survive only when a numerator passes a divisibility test; linked portions of other weights are suppressed to offset the strongest correlated contributions. The rules preserve the required first-moment normalization.

The construction also remembers earlier alignments when intermediate labels disappear. At the first meeting of two previously separate collections, a backward comparison accounts for covariance through entries that passed their tests. A squared-mass bound limits the total contributions assigned to them.

This history-sensitive accounting is the paper’s response to dependencies across many scales. It averages common residue translations, while keeping $\gamma$ fixed throughout; it does not assume independent outcomes or randomize the shift. Once the finite-block estimates are established, the almost-everywhere conclusion follows, and mass transference yields the Hausdorff version.

7. Scope, limitations, and open questions

The theorem does not impose ordinary numerator coprimality. Adding $\gcd(a,q)=1$ changes the statement and, according to the cited counterexamples, can make it false.

Convergence of the totient-weighted sum is not claimed to force a null approximation set. A simple sufficient null-set criterion comes instead from convergence of the unweighted sum $\sum_q\psi(q)$, by the first Borel–Cantelli lemma and interval-length bounds. The gap between these conditions should not be erased by calling the new result a complete convergence-divergence law.

The result gives no rate or asymptotic formula for the number of successful denominators. It also does not treat an arbitrary changing sequence of shifts $\gamma_q$, nor recover the approximation behavior of a specified exceptional real number.

Further questions include quantitative counting, finer dependence on the tolerance support, and meaningful numerator restrictions adapted to particular shifts.

8. Sources and further reading

Theorem 1.1 gives the Lebesgue-measure statement; Corollary 1.2 and Section 11 give the Hausdorff consequence. The introduction carefully distinguishes unrestricted, reduced and shift-adapted numerators. Sections 5–10 develop the weighted finite-block construction and overlap estimates.

The Ramírez and Beresnevich–Hauke–Velani papers linked in section 3 explain the earlier obstruction and rational-shift theorem. The cited counterexample preprints explain why the stronger coprime formulation must remain separate. Optional background on the Duffin–Schaeffer conjecture and Hausdorff measure provides additional context.

Source manuscripts

• M1. OpenAI, The Weak Inhomogeneous Duffin–Schaeffer Conjecture (September 25, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.