Rachel Greenfeld and I have just uploaded to the arXiv our paper “A counterexample to the periodic tiling conjecture“. This is the full version of the result I announced on this blog a few months ago, in which we disprove the *periodic tiling conjecture* of Grünbaum-Shephard and Lagarias-Wang. The paper took a little longer than expected to finish, due to a technical issue that we did not realize at the time of the announcement that required a workaround.

In more detail: the original strategy, as described in the announcement, was to build a “tiling language” that was capable of encoding a certain “-adic Sudoku puzzle”, and then show that the latter type of puzzle had only non-periodic solutions if was a sufficiently large prime. As it turns out, the second half of this strategy worked out, but there was an issue in the first part: our tiling language was able (using -group-valued functions) to encode arbitrary boolean relationships between boolean functions, and was also able (using -valued functions) to encode “clock” functions such as that were part of our -adic Sudoku puzzle, but we were not able to make these two types of functions “talk” to each other in the way that was needed to encode the -adic Sudoku puzzle (the basic problem being that if is a finite abelian -group then there are no non-trivial subgroups of that are not contained in or trivial in the direction). As a consequence, we had to replace our “-adic Sudoku puzzle” by a “-adic Sudoku puzzle” which basically amounts to replacing the prime by a sufficiently large power of (we believe will suffice). This solved the encoding issue, but the analysis of the -adic Sudoku puzzles was a little bit more complicated than the -adic case, for the following reason. The following is a nice exercise in analysis:

Theorem 1 (Linearity in three directions implies full linearity)Let be a smooth function which is affine-linear on every horizontal line, diagonal (line of slope ), and anti-diagonal (line of slope ). In other words, for any , the functions , , and are each affine functions on . Then is an affine function on .

Indeed, the property of being affine in three directions shows that the quadratic form associated to the Hessian at any given point vanishes at , , and , and thus must vanish everywhere. In fact the smoothness hypothesis is not necessary; we leave this as an exercise to the interested reader. The same statement turns out to be true if one replaces with the cyclic group as long as is odd; this is the key for us to showing that our -adic Sudoku puzzles have an (approximate) two-dimensional affine structure, which on further analysis can then be used to show that it is in fact non-periodic. However, it turns out that the corresponding claim for cyclic groups can fail when is a sufficiently large power of ! In fact the general form of functions that are affine on every horizontal line, diagonal, and anti-diagonal takes the form

for some integer coefficients . This additional “pseudo-affine” term causes some additional technical complications but ultimately turns out to be manageable.During the writing process we also discovered that the encoding part of the proof becomes more modular and conceptual once one introduces two new definitions, that of an “expressible property” and a “weakly expressible property”. These concepts are somewhat analogous to that of sentences and sentences in the arithmetic hierarchy, or to algebraic sets and semi-algebraic sets in real algebraic geometry. Roughly speaking, an expressible property is a property of a tuple of functions , from an abelian group to finite abelian groups , such that the property can be expressed in terms of one or more tiling equations on the graph

For instance, the property that two functions differ by a constant can be expressed in terms of the tiling equation (the vertical line test), as well as where is the diagonal subgroup of . A weakly expressible property is an existential quantification of some expressible property , so that a tuple of functions obeys the property if and only if there exists an extension of this tuple by some additional functions that obey the property . It turns out that weakly expressible properties are closed under a number of useful operations, and allow us to easily construct quite complicated weakly expressible properties out of a “library” of simple weakly expressible properties, much as a complex computer program can be constructed out of simple library routines. In particular we will be able to “program” our Sudoku puzzle as a weakly expressible property.

## 43 comments

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30 November, 2022 at 3:31 am

AnonymousVery, very excellent. Thanks you very much

30 November, 2022 at 4:08 am

AnonymousCongratulations on this great result. I think though that the possibility of “issues” is a good reason for avoiding “announcements” altogether.

30 November, 2022 at 6:02 am

Anonymousbut how would that help his coauthor get a job?

30 November, 2022 at 9:49 am

Anonymous994Why the sarcasm? What’s the matter with pushing your postdocs in the job market? Of course that’s what the announcement was all about (otherwise it makes very little sense).

But why is that supposed to be a bad thing?

30 November, 2022 at 12:15 pm

AnonymousQ: “But why is that supposed to be a bad thing?”

A: “The paper took a little longer than expected to finish, due to a technical issue that we did not realize at the time of the announcement that required a workaround.”

30 November, 2022 at 12:41 pm

AnonymousAnd so what? See reply below.

30 November, 2022 at 9:59 am

AnonymousDisagreed! There are two scenarios:

1. The strategy did not work. Then eventually Terry was forced to explain here what went wrong and that is still useful contribution to mathematics (of an underrated kind), which would otherwise remain in some inner circle.

2. The strategy (perhaps with significant fixes) works out. Then there is nothing bad in making some advertising! Especially if you want to promote your postdoc that is on the job market.

If one complains that this is unfair pushing of postdocs: of course at any place where RG’s file has been examined they were well aware that 1 was still possible.

1 December, 2022 at 10:17 am

AnonymousWhy is this has anything to do with his postdoc? He announces all his research on the blog.

1 December, 2022 at 11:36 am

No periodic tiling proof – The nth Root[…] Rachel Greenfeld and I have just uploaded to the arXiv our paper “A counterexample to the periodic tiling conjecture“. This is the full version of the result I announced on this blog a few months ago, in which we disprove the periodic tiling conjecture of Grünbaum-Shephard and Lagarias-Wang. The paper took a little longer than expected to finish, due to a technical issue that we did not realize at the time of the announcement that required a workaround. … (Terence Tao) […]

1 December, 2022 at 1:33 pm

Gil KalaiGreat news! Congratulations Rachel and Terry!

1 December, 2022 at 2:20 pm

AnonymousThis seems to be the most sophisticated mathematical application of a Soduko-like combinatorial design!

5 December, 2022 at 9:22 am

REINALDO F CRISTOCongratulations, the explanatory tags inside the paper are great, keep using this feature widely!

6 December, 2022 at 8:37 am

Apoorva KhareA few “FYI remarks”, re: (the opening) Theorem 1 in the blogpost: it appears that

(1) The test set can be reduced, in that in addition to all (anti)diagonals, one doesn’t need to use *every* horizontal line – just the X-axis will do. Or any one line through the origin, with slope not 1 or -1.

(2) This result – with the reduced test set – extends to all modules over all unital commutative rings with sufficiently large characteristic. (E.g. for “the above setting” of modules with 2 generators, 1+1 should not be a zerodivisor in the ring.)

The proof, including a quantitative refinement (which uses $B_h$-sets / Sidon sets studied by Bose-Chowla and Erdos-Turan), can be found in https://arxiv.org/abs/2212.02429.

6 December, 2022 at 2:54 pm

YahyaAA1You tell a good story – I loved the journey! Signposts along the way point to new ideas that you’ve formulated to help you reach the destination. And seeing how and when you do so can give readers some insight into why maths has so many definitions of [multiply [adjectivally qualified]] objects; e.g., “weakly expressible property” – not a notion that would occur _naturally_ to a student!

16 December, 2022 at 12:30 am

AnonymousQuanta article: https://www.quantamagazine.org/nasty-geometry-breaks-decades-old-tiling-conjecture-20221215/

21 December, 2022 at 4:59 pm

YahyaAA1Thanks for sharing this interesting explanation for non-specialists! And it’s lovely to see Terry playing with his kids’ toys! Mathematicians never DO grow up – thankfully ;-)

22 January, 2023 at 12:53 am

TKAnother conjecture that is apparently false, is the RH. I know that this is extremely unlikely, but if you’re well-versed in basic complex analysis and basic analytic number theory, i kindly invite you to judge this 4-page paper for yourself:

https://figshare.com/articles/preprint/Untitled_Item/14776146

24 January, 2023 at 9:28 am

MuonUnfortunately, as good as this paper evidently is, it will be very hard for it to be taken seriously by the “experts”. By now, we all know that research math operates in mafia style.

24 January, 2023 at 12:42 pm

AnonymousThe standard proceedure is submitting to a top journal. If the author did so, i’m pretty sure some experts are seriously looking at the paper.

But yes, it’s quite remarkable paper IMHO. Credit where it’s due.

24 January, 2023 at 11:24 am

Anonymoussock pupetry at its finest from TK

24 January, 2023 at 3:13 pm

AnonymousWell, it’s a 4-page paper claiming to solve the most important math problem of our time. It’s therefore not too unreasonable for people to be skeptical, nomatter how clear/elegant the argument is. However, i can reliably tell you that a few experts are already taking this paper seriously.

24 January, 2023 at 8:44 pm

AnonymousWe have heard this from TK for many years now – the so-called experts that take him seriously never materialized before and frankly if there were such, why waste time posting here to essentially spam the best math blog there is.

24 January, 2023 at 10:39 pm

TKI have indeed worked very, very hard on the RH for many years, working under extremely difficult socioeconomic conditions, because it’s a problem that i’m deeply passionate about. I made lots and lots of mistakes in the beginning, but in the right direction i would say. The said mistakes encouraged me to work much harder and in the process, i understood the problem better.

It’s easy for you to misinterpret my posting here as “spamming”, because you have absolutely no idea of what it’s like to be in my shoes. It took a lot of sacrifice and focus forme to reach this point. We do not have any analytic number theorist in this part of the world, whereas this blog it’s a place that is frequently visited by expert analytic number theorists, some of whom are open-minded and kind-hearted enough to evaluate my paper solely on merit. That’s the reason why I’m posting here. I’m not “spamming” !

Since I’m only interested in math, i won’t respond to these type of comments any further.

Regards,

TK.

25 January, 2023 at 3:25 am

AnonymousIn this lecture:

Maynard mentions (between the 15th and 20th minutes) that there could be some zeros of zeta on the 3/4 line. Curiously, TK’s paper claims that the supremum of the real parts of the zeta zeros is at least 3/4.

26 January, 2023 at 8:09 pm

DaveMathematics should make sense!

A Prime Example: We accept or assume the Riemann Hypothesis (RH) is true since the truth of RH makes sense…

For those who think RH is false, we demand proof:

1. Find at least one counterexample that demonstrates RH is false.

2. Explain why any counterexample to RH makes sense.

3. Otherwise, stop promoting fantasies or more nonsense since they damage mathematics generally.

Dave.

27 January, 2023 at 8:28 am

TKFirstly, the statement that the “RH makes sense”, is quite vague. A more accurate statement is that, “the RH is highly likely to be true, given numerical and heuristic evidence.”

Concerning your “points” 1-3, I will address them in the order you numbered them.

1.) To disprove the RH, it’s not necessary to produce an explicit counterexample – a non-constructive argument indeed suffices. In fact, for number-theoretic purposes, we are more interested in knowing lower bounds for the supremum of the real parts of the zeta zeros, rather than the actual zero off the critical line. The former is what improves, for instance, our existing knowledge of how large the error term of the PNT can be.

2.) There are plenty of (well-known) reasons that explain why the falsehood of the RH would make sense. Read, for example, Aleksander Ivic’s article: “Some reasons to doubt the RH”.

Also, as Mayanard mentioned in the lecture linked above, there are reasons to believe that there could be some zeros of zeta on the 3/4 line. This is precisely in agreement with Corollary 1 of my paper! I should mention that i wasn’t even aware of Maynard’s lecture until now, otherwise I would have cited the results he mentioned, as they can be viewed as solid heuristic evidence why my Corollary 1 should be true.

Last but not least, Rodgers and Tao demonstrated that the de Bruijn-Newman constant is nonnegative. This OFFICIALLY proved that if the RH is true, then it is just barely so.

3.) I’m not promoting “fantasies”. Rather, I’m simply sharing my research on a platform several experts meet, some of whom are kind and open hearted enough to factually judge my paper on merit. I have worked extremely hard on this, and i’ll certainly not be demotivated by ignorant and factless comments like yours.

It’s very clear from your comment that you’re totally incompetent in basic analytic number theory, so it’s pointless to respond to you any further.

-TK.

27 January, 2023 at 9:21 pm

Dave“Science is an adventure of the human spirit. It is essentially an artistic enterprise, stimulated largely by curiosity, served largely by disciplined imagination, and based largely on faith in the reasonableness (making sense), order, and beauty of the universe.”

— Mathematician/Scientist Warren Weaver.

27 January, 2023 at 9:28 pm

DaveFYI: Number Theory Forum @ https://www.math10.com/forum/viewforum.php?f=63

27 January, 2023 at 11:18 pm

AnonymousDave, it’s quite clear that you’re not a research mathematician. Please, stick to philosophy.

27 January, 2023 at 10:48 pm

AnonymousA lot of things in mathematics that appear to “make sense”, are not necessarily true. That’s why we rely on rigorous proofs.

27 January, 2023 at 11:21 pm

DaveMath that make sense means clearness, soundness (via proofs, computations, etc.), reasonableness, correctness, etc.

Either something makes sense or it is nonsense.

If the math (e.g. RH) makes sense, then it is sound/true/clear/reasonable. In the case of RH, there is a deep reason (or there may be some deep reasons) behind all numerical evidence that supports it.

Example of a Possible Deep Reason for RH: Is the critical line = 1/2 optimal?

1 February, 2023 at 12:50 pm

Anonymous“Is the critical line = 1/2 optimal?” Yes!! 👍😃👍

25 January, 2023 at 8:33 am

Nori KermicheThis could be a nice example for Terry’s Oscillatory Integrals Lectures. The article uses both stationary and non-stationary phase methods to approximate of the zeta function in the critical strip.

https://www.techrxiv.org/articles/preprint/The_Riemann_Zeta_function_using_the_Method_of_Stationary_Phase_Approximation_of_the_Van_der_Pol_Fourier_Integral/21938345

27 January, 2023 at 11:58 pm

DaveThanks for the personal attacks, mischaracterizations, etc.!

The Riemann Hypothesis is true! 👍✌

Go Blue! 👍✌

28 January, 2023 at 12:31 am

AnonymousNo one ever attacked you personally. And please, stop using a VPN (or whatever you’re using) to upvote your nonsensical posts and downvoting every other sensible post.

In as much as you have upvoted your nonsensical comments, they remain void of any facts. And in as much as you have downvoted the other comments (e.g. the one with the link to Maynard’s lecture), they remain sensible.

Unfortunately for you, math does not work with votes, but facts.

30 January, 2023 at 7:19 pm

AnonymousHas the work been published officially? Which journal?

31 January, 2023 at 12:22 am

AnonymousIn a year or two you’ll see it in Annals, JAMS, or Acta.

31 January, 2023 at 9:20 am

AnonymousHeard this for 10+ years and it’s always next year…

31 January, 2023 at 9:29 am

AnonymousAre you slow? This paper is a few weeks old.

31 January, 2023 at 11:40 am

Anonymousof course, this particular version is a few weeks old, but it is version who knows what, 40, 80 whatever, and is fatally flawed like the others; I predict that in a few months, we have the next version proving RH again rather than disproving it like here

3 February, 2023 at 11:24 am

AnonymousIt depends on how you look at it. From another angle, one would actually respect the author for the focus & hard work they put into teaching themselves under very difficult conditions, until they finally disproved RH. And yes, take it from me, the author indeed disproved RH.

3 February, 2023 at 2:32 pm

AnonymousA Fundamental Rule of Prime Number Theory:

For ALL integers, k > 1, there exists a prime number, p, which DIVIDES k such that either p = k or p is less than or equal to the square root of k.

BIG HINT: The square root (the OPTIMAL exponent, 1/2) of k is confirmed by the Riemann Hypothesis.

FYI: Riemann Hypothesis (Re(z) = a = 1/2) for ALL nontrivial zeros, z = a ± bi, of the Riemann Zeta Function (RZF).

“Nothing takes place in the world whose meaning is not that of some maximum or minimum.” ― Leonhard Euler.

Important Reminder: Nontrivial zeros of RZF detect prime numbers…

1 February, 2023 at 12:56 pm

Dave“Is the critical line = 1/2 optimal?” Yes! 😃