How well can fractions approximate pi?

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Family 017 · Number theory
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

The fractions $22/7$ and $355/113$ approximate $\pi$ surprisingly well for their small denominators. But isolated successes do not tell us how good rational approximations can remain as the denominator grows. The real question is whether there are infinitely many exceptionally accurate fractions.

Family 017 claims the sharp answer: the irrationality exponent of $\pi$ is exactly two. Roughly, errors of order the inverse square of the denominator occur infinitely often, but improving the exponent by any fixed positive amount eventually becomes impossible.

This also answers a question that initially looks unrelated. Does the positive series

$$\sum_{n=1}^{\infty}\frac{1}{n^3\sin^2 n}$$

converge when angles are measured in radians? The factor $n^{-3}$ encourages convergence, but an integer very close to a multiple of $\pi$ makes the sine exceptionally small. The manuscript claims that its approximation theorem controls these rare spikes well enough to prove convergence.

2. Background and essential definitions

For an irrational real number $x$, its irrationality exponent is

$$\mu(x)=\sup\left\{\nu>0: 0<\left|x-\frac pq\right|<q^{-\nu} \text{ for infinitely many coprime }p,q,\ q\ge2\right\}.$$

A larger exponent means that exceptionally accurate rational approximations occur more often in this asymptotic sense. The definition concerns infinitely many fractions, not one unusually good approximation.

Dirichlet’s approximation theorem, obtainable from the pigeonhole principle, implies $\mu(x)\ge2$ for every irrational $x$. Continued fractions give an explicit sequence of good approximations, the convergents $p_j/q_j$, satisfying

$$\left|x-\frac{p_j}{q_j}\right|<\frac{1}{q_j^2}.$$

Numbers can nevertheless have much larger irrationality exponent. A Liouville number admits approximations beating every fixed power of the denominator. Irrationality by itself therefore gives no finite upper bound.

The statement $\mu(x)=2$ is best read with an epsilon:

$$\text{for every }\varepsilon>0,\quad \left|x-\frac pq\right|\ge q^{-2-\varepsilon} \quad\text{for all sufficiently large }q.$$

The threshold can depend on $\varepsilon$. This is weaker than a fixed inequality $|x-p/q|\ge c/q^2$ with one positive constant $c$. That stronger property is equivalent to bounded partial quotients in the continued fraction expansion. Exponent two does not settle whether those partial quotients are bounded.

For the series application, write

$$\|y\|=\min_{m\in\mathbb Z}|y-m|$$

for distance to the nearest integer. A close approximation $p/q$ to $\pi$ makes $\|q\pi\|$ small. If $p$ is near $q\pi$, then

$$|\sin p|=|\sin(p-q\pi)|\le |p-q\pi|.$$

Thus small approximation errors produce large terms of the sine-denominator series at integer numerators $p$. The arithmetic difficulty is to understand how numerous and how clustered such terms can be.

3. Development history and previous partial results

The irrationality and transcendence of $\pi$ are classical, but neither supplies the predicted exponent two. Transcendence says that $\pi$ satisfies no nonzero polynomial equation with integer coefficients. Rational approximation asks a different quantitative question.

Mahler’s 1953 work gave a finite irrationality measure, with exponent 42 for all denominators at least two and an eventual improvement to 30. Mignotte refined the method to 21 and eventually 20. Hata’s integral constructions reached approximately $8.01604539$, and Salikhov improved this to approximately $7.606308$. The manuscript’s bibliography identifies these original papers; Zudilin’s essay on irrationality measures explains the logarithmic approximation and denominator arithmetic behind this line of work.

Zeilberger and Zudilin subsequently obtained

$$\mu(\pi)\le7.103205334137\ldots$$

through a variant of Salikhov’s integral construction, with rigorous arithmetic and asymptotic analysis. See their 2019 preprint and 2020 publication. The current manuscript also cites Bai’s September 2026 preprint, which claims a further improvement below $7.101862832357$. These earlier numerical bounds provide context; the new argument does not use them as inputs.

The Flint–Hills series has its own history. Alekseyev showed that convergence requires $\mu(\pi)\le5/2$. Meiburg proved the sufficient strict condition $\mu(\pi)<5/2$, as explained in Bounds on Irrationality Measures and the Flint-Hills Series. Those two implications do not settle the equality boundary $\mu(\pi)=5/2$.

Against this background, the claim $\mu(\pi)=2$ is a qualitative jump to the predicted optimum, rather than another small numerical improvement. It lies comfortably inside the known sufficient range for convergence of the classical series.

4. What the new manuscripts claim

The family contains one manuscript. Its Theorem 1.1 states

$$\boxed{\mu(\pi)=2.}$$

More explicitly, for every real $\nu>2$, there exists an integer $Q(\nu)$ such that

$$\left|\pi-\frac pq\right|\ge q^{-\nu} \qquad(p\in\mathbb Z,\ q\ge Q(\nu)).$$

This inequality applies to all integer numerators and denominators, whether or not the fraction is reduced. The argument does not give an effective value of $Q(\nu)$.

Corollary 1.2 claims convergence of the classical Flint–Hills series. Corollary 5.2 gives a broader exact criterion: for fixed real $a,b>0$,

$$\sum_{n=1}^{\infty}\frac{1}{n^a|\sin n|^b} \quad\text{converges if and only if}\quad a>\max\{1,b\}.$$

All angles are in radians. Both boundary lines are excluded from convergence. If $a\le1$, the terms dominate those of the divergent series $\sum n^{-a}$. If $1<a\le b$, continued-fraction numerators give a subsequence whose summands fail to approach zero. The new approximation theorem supplies the convergence direction in the remaining region.

For instance, $a=3,b=2$ gives the classical convergent case. The case $a=b=2$ diverges, despite the familiar convergence of $\sum n^{-2}$, because unusually small sine values keep a subsequence of terms large.

5. Why the advance matters

Provisional importance: 4/5. Taken as stated, this resolves a prominent sharp approximation problem for a fundamental transcendental constant and yields a clean boundary for an entire family of singular trigonometric series.

The optimal value two is the benchmark supplied by elementary approximation from below. Reaching it from an upper bound above seven requires controlling all hypothetical infinite sequences of exceptional approximations, not merely optimizing a familiar integral.

The method also introduces a multivariable interpolation statement whose thresholds are independent of the interpolation centers. That kind of uniformity could matter beyond this particular constant. The rating reflects the central approximation problem and methodological reach; the statement remains specifically about $\pi$, not all familiar transcendental numbers.

6. The main idea

Construct an object that cannot be too small

A standard irrationality strategy compares arithmetic discreteness with analytic smallness. A nonzero integer has absolute value at least one. More generally, a nonzero rational number with controlled denominator has a known lower bound. If analysis forces the same quantity below that bound, the assumed approximation was impossible.

Here the object is a determinant with entries in $\mathbb Q(i)$, where $i^2=-1$. Clearing denominators gives an arithmetic lower bound for a nonzero minor. The challenge is to ensure nonvanishing while exploiting several exceptionally accurate approximations at once.

Assume, for contradiction, that one fixed $\nu>2$ admits infinitely many approximations beating $q^{-\nu}$. The manuscript chooses finitely many with successively separated denominators. Rational multiples of these fractions approximate the exact exponential periods $2\pi i j$, for integers $j$.

Interpolate along logarithmic curves

A jet is a finite collection of Taylor coefficients. The interpolation theorem prescribes jets of a polynomial after substitutions of the form

$$Y=1+t,\qquad X_i=c_{ji}+u_i+\log(1+t).$$

The variable $t$ follows a logarithmic curve, while $u_i$ measures transverse displacement. The centers $c_{ji}$ come from the chosen rational approximations. Weighted degree bounds keep only polynomial monomials whose weighted exponent sum fits inside a simplex.

Merely having more coefficients than conditions does not establish that arbitrary jets can be prescribed: the special positions of the centers might create hidden dependencies. The manuscript’s interpolation theorem establishes the required full rank, using multiplicity estimates, logarithmic residues and geometric positivity.

The order of choices matters. Dimension and successive coordinate weights are chosen before the center locations. The approximating denominators can then be selected to meet those already fixed thresholds. Finally the polynomial degree grows. This prevents the accuracy requirement from depending circularly on the approximations selected to satisfy it.

Two competing sources of analytic smallness

The determinant rows are translated from approximate periods to exact periods and expanded in the transverse variables. Large transverse indices contribute many powers of the tiny rational-approximation errors.

If many rows instead have small transverse indices, they fall into relatively few groups. Rows in the same group test Taylor coefficients of the same entire functions. Repeated Taylor degrees then make a determinant term vanish, so surviving terms must use sufficiently distinct degrees. This produces another strong saving.

The argument therefore covers both possibilities: many large transverse indices supply small errors, while many small indices force Taylor-degree separation. Full weighted simplices control the coefficient count and denominator cost together. Increasing the number of variables makes the combined analytic saving contradict the arithmetic lower bound for every fixed $\nu>2$.

Why the series needs spacing, not only individual bounds

Choose $2<\nu<5/2$. The theorem yields a positive constant $c$ with

$$\|q\pi\|\ge c q^{1-\nu}\qquad(q\ge1),$$

after absorbing finitely many small denominators into $c$. For $K\le q<2K$, the points $q\pi$ modulo one are separated from zero and from each other by a distance comparable to $K^{1-\nu}$. The same inequality applied to differences of denominators gives the mutual separation.

Ordering these distances on the circle bounds a whole dyadic block:

$$\sum_{K\le q<2K}\frac{1}{q^3\|q\pi\|^2} =O(K^{2\nu-5}).$$

Because $2\nu-5<0$, summing over $K=1,2,4,\ldots$ converges. Grouping integers near successive multiples of $\pi$ transfers this estimate to the original sine series. This controls the frequency of spikes as well as their possible size; a crude separate estimate for every term would lose that information.

7. Scope, limitations, and open questions

The manuscript does not claim bounded continued-fraction partial quotients of $\pi$, a uniform positive constant at exponent exactly two, or an effective denominator threshold. Those are stronger or different conclusions.

It also gives no claim about normality of the decimal expansion of $\pi$, digit frequencies or statistical independence of its digits. Rational approximation and digit randomness are distinct questions. Nor does the result automatically give exponent two for unrelated constants.

The generalized series criterion concerns fixed positive $a,b$ and radian angles. It gives convergence and divergence, not an effective error bound for numerically summing the series. Exceptionally large terms can make numerical experiments misleading even when a positive series converges.

8. Sources and further reading

Theorem 1.1 is the main approximation claim. Theorem 2.1 states the separated-weight interpolation result; Sections 3–4 compare determinant estimates. Section 5 contains the dyadic spacing argument and Corollary 5.2’s exact generalized-series criterion.

The primary papers linked in section 3 trace the previous irrationality bounds and the Flint–Hills threshold. For optional background, continued fractions explain the convergents used above, and irrationality measure compares exponent two with stronger approximation properties.

Source manuscripts

• M1. OpenAI, The irrationality exponent of π is 2 (September 24, 2026).

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