# Nontrivially fillable gaps in published proofs of major theorems

Prelude: In 1998, Robert Solovay wrote an email to John Nash to communicate an error that he detected in the proof of the Nash embedding theorem, as presented in Nash's well-known paper "The Imbedding Problem for Riemannian Manifolds" (Annals of Math, 1956), and to offer a nontrivial fix for the problem, as detailed in this erratum note prepared by John Nash. This topic is also discussed in this MO question.

Of course, any mathematician who has been around long enough knows of many published proofs with significant gaps, some provably irreparable, and some perhaps authored by himself or herself. What makes the above situation striking--and discomforting to many of us--is the combination of the following three factors:

(1) The theorem whose proof is found faulty is a major result that was published in 1950 or after, in a readily accessible source to experts in the field . (I chose the 1950 lower bound as a way of focusing on the somewhat recent past).

(2) The gap detected is filled with a nontrivial fix that is publicly available and consented to by experts in the field (so we are not talking about gaps easily filled, or about gaps alleged by pseudomathematicians, or about false publicly accepted theorems, as discussed in this MO question).

(3) There is an interlude of 30 years or more between the publication of the proof and the detection of the gap (I chose 30 years since it is approximately the age difference between consecutive generations, even though the interlude is 42 years in the case of the Nash embedding theorem).

Question to fellow mathematicians: what is the most dramatic instance you know of where all of the three above factors are present?

• I caught an error in the proof of the main result of arxiv.org/abs/1309.4516 It's been a while, but I seem to recall that Prop 3.5 is the invalid culprit. I didn't say anything at first, because my PhD advisor told me it would be inappropriate to do so (???) and I deferred to his judgement. But after it passed peer review and was published, I finally pointed out the error to one of the authors. He agreed that the proof was invalid, but insisted the conclusion is true. It's not a huge deal I guess, but sorta disheartening to know that mathematicians won't admit their errors publicly. – Ben W Feb 9 at 2:52
• "Dulac's theorem" on the finiteness of the number of limit cycles of a two-dimensional polynomial system is an extreme example of difficulty in "filling the gap", but it does not qualify since Dulac's "theorem" was published in the 1920s. – Alexandre Eremenko Feb 9 at 14:15
• @BenW The advice given to you by your former advisor is very unfortunate and rather atypical of the attitude of most mathematicians I know of. – Ali Enayat Feb 9 at 23:00
• This doesn't meet your 30-year threshold but I think it's an interesting example nonetheless: The Fight to Fix Symplectic Geometry. – Timothy Chow Feb 10 at 3:23
• The question reminded me of the four color theorem, for which a couple early proofs had gaps that took 11 years to recognize, but then it took nearly 100 years after that to actually fill the gaps. – Daniel Schepler Feb 12 at 1:16

In 1970, I. N. Baker published a proof of a basic result in holomorphic dynamics:

a transcendental entire function cannot have more than one completely invariant domain.

A completely invariant domain is an open connected set $$D$$ such that $$f(z)\in D$$ if and only if $$z\in D$$.

Baker "proved" a more general statement that: there cannot be two disjoint domains whose preimages are connected.

The "proof" was a simple topological argument which occupied less than one page. Since then this result has been used and generalized by extending his simple argument. In summer 2016 I was explaining Baker's argument to Julien Duval, he was somewhat slow in understanding and kept asking questions. Few weeks later he found a gap in the proof. It also took him some time to convince me that there is a gap indeed. Specialists were informed.

Half a year later an amazing counterexample has been constructed in https://arxiv.org/abs/1801.06359 by Lasse Rempe-Gillen and David Sixsmith. This paper contains the full account of the story. This is a counterexample to Baker's more general statement only, not to the highlighted theorem itself, which is now an important open question.

• (Was any gap filled?) – Francois Ziegler Feb 9 at 5:04
• @Francois Ziegler: No. The counterexample I refer to shows that the gap cannot be filled. Some completely new idea is required. So this example is "NON (trivially-fillable)" rather than "(non-trivially) FILLABLE" :-) – Alexandre Eremenko Feb 9 at 14:34
• So technically this doesn't answer Ali Enayat's question, I guess? It seems to be a requirement that the gap be filled, since Enayat explicitly excludes false publicly accepted theorems. (Mind you, I like your answer and I upvoted it; I'm just pointing out an apparent technicality.) – Timothy Chow Feb 9 at 20:44
• I agree with all comments. But I will not remove my answer since it has many votes:-) – Alexandre Eremenko Feb 10 at 1:44
• @AlexandreEremenko I am all for keeping your answer, since what you reported in your answer deserves wide dissemination. – Ali Enayat Feb 10 at 23:23

In 2017 an erratum to the 1973 paper Isotopies of homeomorphisms of Riemann surfaces by Birman and Hilden appeared in Annals of Mathematics which satisfies your three criterias. That's a 43-year gap! The way Birman and Hilden tracked all papers citing theirs is admirable.

The error was found by Ghaswala, and a fix was provided by Ghaswala and Winarski in Lifting Homeomorphisms and Cyclic Branched Covers of Spheres, published the same year as the erratum.

• Thank you very much for your enlightening answer, it nicely meets the three criteria stipulated in my question. – Ali Enayat Feb 10 at 1:18

If a 25-year interlude will do, there is

R. F. Coleman has sent me his preprint ["Manin's proof of the Mordell conjecture'', Preprint, 1988; per bibl.] concerning my proof of Mordell's conjecture for function fields (see the paper cited in the heading). Coleman has discovered and corrected inaccuracies in my paper. Below I explain what changes should be made in the original paper in the language of that paper.

(If not, then maybe this.)

A lightweight counterpart: a false argument (ascribed to Hilbert) and a false statement by Cauer in 1910s: one cannot find a center of a circle [Hilbert] or two disjoint circles [Cauer] using a ruler only. A wrong argument can be found in most popular books [e.g. Courant/Robbins, or Rademacher/Toeplitz], the error was noted just a few years ago:

Arseniy Akopyan, Roman Fedorov, Two circles and only a straightedge, Proc. AMS 147 no. 1 (2019) pp. 91-102, doi:10.1090/proc/14240, arXiv:1709.02562.

• That's an impressively long gap - over 100 years! – Wojowu Feb 12 at 9:17

In 1980, Micali and Vazirani published An $$O(\sqrt{|V|}\cdot |E|)$$ algorithm for finding maximum matching in general graphs. I regard this as a major result in theoretical computer science. By Vazirani's own account, a complete proof of the running time claimed in the title was not provided until his 2012 arXiv preprint. That is a gap of 32 years.

However, one could object that the 1980 paper was technically just an "extended abstract" that did not claim to provide a full proof of correctness. In 1994, Vazirani published a paper claiming to give a proof (but which, as he himself acknowledged in his 2012 preprint, contained gaps and errors). So the gap is arguably "only" 18 years.

In 1990, Ravi Kannan wrote a paper giving an algorithm deciding $$\forall \exists$$ sentences of integer programs. As an intermediate claim, he "proved" the "Kannan Partition Theorem". Because his proof was unreadable to Eisenbrand and Shmonin who wanted to extend his result, they proved their own slightly weaker version of KPT. In 2017, Nguyen and Pak showed that if the KPT is true, then Short Presburger Arithmetic sentences can be decided in polynomial time, but a few months later, they showed that this is in fact hard and discovered the bug in the proof of KPT. The weaker version of Eisenbrand and Shmonin holds and is sufficient to prove Kannan's original final result.

So here the bug was fixed by Eisenbrand and Shmonin before it was discovered by Nguyen and Pak, and a (conditional) positive result was derived from it by the same authors who (a few months later) disproved it.

In 1979, Dobkin and Snyder published an algorithm that purported to give the largest-area triangle inscribed in a convex n-gon in O(n) time. In 2017, Keikha, Löffler, Urhausen, and van der Hoog showed that this algorithm was in fact wrong. Two different authors were quickly and independently able to fill the gap and give a correct linear-time algorithm (I was one of them). It later turned out that the gap was already filled in a 1992 paper by Chandran and Mount, where they describe a linear-time algorithm to simultaneously construct the largest inscribed and smallest circumscribed triangles. Because the 1979 algorithm was not known to be wrong at the time, the fact that it gave an O(n) algorithm for the largest inscribed triangle was not a selling point of the 1992 paper and was not emphasized.

In 1986, Partha Dasgupta and economic Nobel laureate Eric Maskin published a paper with an incorrect definition of symmetric games in it. The paper has over 1200 citations and I'm pretty sure it took 25 years until I pointed out on Wikipedia in 2011 that there's a problem (see the edits of this page). The problem is outlined in this with more work on the topic in this more recent draft (currently developing software though).

• Could you clarify why "incorrect definition" counts as "gaps in published proofs"? – Yemon Choi Feb 19 at 20:43
• It has had over 1200 citations since, of which have citations themselves, that can easily lead to all sorts of gaps, and seems like a noteworthy gap in and of itself, though yes is not for a proof itself. – Nicholas Ham Feb 29 at 7:38

I think that the formula for the Littlewood-Richardson rule (how to expand a product of Schur functions into Schur functions) qualifies. It was first claimed to be proved 1934, then an error was discovered and fixed in 1938. The first complete proof was given in 1977, and nowadays there are many different short proofs.

• I seem to remember somebody (possibly Gian-Carlo Rota) summarizing this situation by saying something like "Putting men on the moon was easier than proving the Littlewood-Richardson rule". – Robert Furber Mar 1 at 7:33
• "Unfortunately the Littlewood–Richardson rule is much harder to prove than was at first suspected. The author was once told that the Littlewood–Richardson rule helped to get men on the moon but was not proved until after they got there." --Gordon James (1987) – Per Alexandersson Mar 1 at 7:44