Great theorems with elementary statements: 2026-onward

Question

My 2021 book

Landscape of 21st Century Mathematics, Selected Advances, 2001–2020

collects great theorems with elementary statements published in 2001-2020. I now finishing the second edition of this book (draft available on request, e-mail [email protected]), which will cover the period 2001-2025.

I would like this mathoverflow question to complement the book and display the big list of great theorems with elementary statements to be published from 2026 onward.

Here, by "great" I mean on the level of top math journals, such as Annals of Mathematics, Journal of AMS, Acta Mathematica, Forum of Math Pi, or Inventiones mathematicae. I do not require the paper to be published in these journals, just be on the same level (or above). For example, Perelman's proof of the Poincare conjecture exists as arXiv preprint only but is obviously great.

"Elementary statement" means that you can rigorously formulate the result here, starting with only basic undergraduate background. If the main result of the corresponding paper is too technical, but you can present its easier-to-state special case, or elementary corollary, this would be especially interesting. It would be nice if you also provide some background and context, also on elementary level.

I am asking this question now because all top journals just finished the publications of their last issues of 2025. So, all papers not currently published (accepted for publications, under review, just appered in arXiv, etc.) are expected to be published from 2026 onward, and therefore qualify.

It is ok to answer with your own theorems, as soon as they satisfy the requirements. In fact, in cases when the main result is too technical, the author may be in best position to state an elementary reformulation or corollary.

• Kimball: What is above the level of Annals, JAMS, etc? Also, are you asking for all such theorems to be published from 2026-$\infty$? Wouldn't it make more sense to ask a question about the future in the future?
• Timothy Chow: The Stanley-Stembridge conjecture, now Hikita's theorem, qualifies. While I have no inside information, my guess is that it will be published in JAMS.
• Sam Hopkins: I think Jineon Baek's solution of the Moving Sofa Problem is considered correct by the experts and this paper will be published in 2026 or beyond; see also this Quanta article about the solution.
• Timothy Chow: @SamHopkins I like that result as well, but I have doubts about whether it is "great" in the sense that Bogdan Grechuk has defined the term. On the other hand, it seems to me that many of the results in Grechuk's 2021 book also fail to be "great" in this sense.
• Bogdan Grechuk: About 90 percent of theorems described in my book are from the listed top journals. I also included some theorems based on other criteria, such as (i) win of major prize that is given for a specific paper, or (ii) very high citation comparing to other papers published in the same year and same subject category, etc. However, for future papers it is difficult to predict prizes or citations, so I just define "great" as "top-journal-level" for the purpose of this mathoverflow question.

Answer 1

To my mind the interesting and highly nontrivial yet stated in elementary terms result, which deserves to appear in Your Book, is the Halpern-Weaver conjecture, which was resolved by Richard Evan Schwartz and published in Annals of Mathematics in 2025.

DOI: https://doi.org/10.4007/annals.2025.201.1.5

It is well known that one is able to make a Moebius band having a sufficiently long strip 1ドル \times \lambda$. The paper of Richard Schwartz answer the following question:

"What is the minimal value of ratio $\lambda$, so that it is possible to make the Moebius band of a 1ドル\times \lambda$ paper strip applying only smooth isometric transformations to it?".

And the result present in the paper asserts that $\lambda > \sqrt{3}$ and this bound is optimal.

• Bogdan Grechuk: Completely agree that this result is great, has elementary statement, and is perfect for my book. I have already included it in the current draft.

Answer 2

In a preprint posted online in 2025, Wang and Zahl proved the Kakeya conjecture in ${\mathbb R}^3$. The paper is currently under review and will be published in 2026+.

A Kakeya set in ${\mathbb R}^3$ is a compact subset of ${\mathbb R}^3$ which contains a unit line segment in each direction. It is known that such set can have Lebesgue measure 0ドル$. A conjecture known as the Kakeya conjecture in ${\mathbb R}^3$ predicted that every Kakeya set $S$ in ${\mathbb R}^3$ has Hausdorff dimension $\dim_H(S)$ equal to 3ドル$.

Davies proved ${\mathbb R}^2$ version of the conjecture, which implies that every Kakeya set $S$ in ${\mathbb R}^3$ has Hausdorff dimension $\dim_H(S)$ at least 2ドル$. In 1991, Bourgain improved the lower bound to $\dim_H(S)\geq \frac{7}{3}$. In 1995, Wolff improved it further to $\dim_H(S)\geq\frac{5}{2}$. The Wolff's bound remained the best known for over 20 years. In 2019, Katz and Zahl improved it to $\dim_H(S)\geq\frac{5}{2}+\epsilon_0$ for some $\epsilon_0>0$. While this improvement is very small, the importance of this result lies in the fact that there are sets of Hausdorff dimension 5ドル/2$ which closely resemble Kakeya sets, and Katz and Zahl demonstrated, for the first time, how to deal with this difficulty. After this, there was a hope for more rapid further progress, and this is what indeed happened.

In a talk in 2014, Tao described an approach for solving the Kakeya conjecture in ${\mathbb R}^3$, which became known as the Katz-Tao program. At the heart of the program is the notion of sticky Kakeya set, defined below. For every (affine) line $\ell \subset {\mathbb R}^3$, let $p=p(\ell)$ be the (unique) point on $\ell$ such that $p \perp \ell$. Then set $\mathcal{L}$ of all lines in $\mathbb{R}^3$ is a metric space with distance
$$ \rho(\ell, \ell') := |p(\ell) - p(\ell')| + \angle(\ell, \ell'). $$ A compact set $S \subset \mathbb{R}^3$ is called a sticky Kakeya set if there exists a set of lines $L \subset \mathcal{L}$ with packing dimension 2ドル$ that contains at least one line in each direction, such that for every $\ell \in L$ the intersection $\ell \cap S$ contains a unit interval. Note that this is exactly the definition of an ordinary Kakeya set, with the additional requirement that $L$ has packing dimension 2ドル$. The Katz-Tao program suggested to first (i) show that any potential counterexample, that is, any Kakeya set in ${\mathbb R}^3$ with Hausdorff dimension $d<3$, must be sticky, and also must have two other structural properties called planiness and graininess, and (ii) use these properties to show that such a counterexample cannot exist.

In fact, planiness was already known. In 2016, Guth proved graininess, so it remained to prove stickiness to complete part (i) of the Katz-Tao program. In 2025, Wang and Zahl implemented this program in full. First, part (ii) of the program was completed by showing that a sticky counterexample cannot exist.

Theorem 1. Every sticky Kakeya set in ${\mathbb R}^3$ has Hausdorff dimension 3ドル$.

Then Wang and Zahl posted online a preprint in which they proved that any Kakeya set in ${\mathbb R}^3$ with the Hausdorff dimension $d<3$ must be sticky and therefore cannot exist by Theorem 1, thus establishing the full Kakeya conjecture in ${\mathbb R}^3$

Theorem 2. Every Kakeya set in ${\mathbb R}^3$ has Hausdorff dimension 3ドル$.

• Francois Ziegler: What is "$p$" where you first use it ("$p \perp \ell$")?
• Bogdan Grechuk: $p=p(\ell)$ is the point on the line, such that vector from the coordinate center to $p$ is orthogonal to $\ell$.

Answer 3

The so-called ''Eremenko's Conjecture'' (1989) is very easy to state: for an entire function $f$, denote by $f^{(n)}$ the $n$-th iterate, and consider the set $$I(f)=\{ z:f^{(n)}(z)\to\infty,\; n\to\infty\}.$$ This is called the escaping set. In 1989 I proved that it is always non-empty (if $f$ is not a polynomial of degree $\leq 1$), and conjectured that every connected component of $I(f)$ is unbounded. There was a lot of research on this question, and it was considered a central unsolved problem in dynamics of transcendental entire functions.

A counterexample was finally found in the paper

David Martí-Pete, Lasse Rempe, and James Waterman, Eremenko’s conjecture, wandering lakes of Wada, and maverick points.Zbl 08062101 J. Am. Math. Soc. 38, No. 4, 877-918 (2025).

• Sam Hopkins: This paper looks like it was published in 2025, so outside the requested timeframe.
• Alexandre Eremenko: But today it is still 2025. How can we say anything about $\geq 2026$??
• Sam Hopkins: I guess Bogdan Grechuk is looking for papers that are in press at top journals, slated to be published next year and beyond (but I didn't write the question, and in general don't really like these kind of questions anyways).
• Alexandre Eremenko: The latest edition of the book he refers to mentions the conjecture but does not mention that it is solved. So I wanted to inform the author:-)
• Bogdan Grechuk: @Alexandre Eremenko. Thank you. This theorem indeed fits to the period 2001-2025 to be covered in the second edition of my book. It is great and easy to state, so it will 100% be included. On the other hand, there are a lot of great papers that are currently accepted for publications, or under review, or just submitted to arxiv. These papers will be published in 2026 or later. The paper "Universal Plücker coordinates for the Wronski map and positivity in real Schubert calculus" from your other answer is an example.

Answer 4

Grassmannian $G(m,n)$ is the set of $m$-dimensional planes in $C^n$. It can be represented as the set of $m\times n$ matrices with at least one $m\times m$ minor is $\neq 0$, modulo multiplication from the left on non-singular $m\times m$ matrices. Another way to think of it is the set of $m$-dimensional subspaces in the space of polynomials of degree $n-1$ in one variable.

Taking the Wronskian determinant of a basis in such a subspace of polynomials defines the "Wronski map" $$G(m,n)\to P^{m(n-m)}$$ where $P^{m(n-m)}=G(1,m(n-m)+1)$ is the projective space, identified with the set of non-zero polynomials of degree $\leq m(n-m)$, modulo proportionality.

The positive Grassmannian consists of those elements of $G(m,n)$ which can be represented by matrices where all $m\times m$ minors are positive.

The theorem I am talking about says that if $p$ is a polynomial with all negative roots, then all elements of the preimage of $p$ under the Wronski map belong to the positive Grassmannian.

This is proved in a paper by Steven Karp and Kevin Purbhoo, "Universal Plücker coordinates for the Wronski map and positivity in real Schubert calculus".

This is a very broad and unexpected generalization of the former B. and M. Shapiro conjecture, and the conjecture discussed in this MO post. It also contains a proof of several other conjectures in real algebraic geometry.

I also mention that positive Grassmannians play an important role in mathematical physics besides combinatorics and algebraic geometry.

Answer 5

Markoff triples are positive integers solutions $(x,y,z)$ to the equation \begin{equation}\label{eq:Markoff} x^2+y^2+z^2=3xyz, \end{equation} which is known as Markoff equation. Any integer that appear in any of the Markoff triples is called Markoff number. It is known and easy to see that all Markoff triples can be generated from $(1,1,1)$ by Vieta jumping operations $$ (x,y,z) \to (3yz-x,y,z), ,円,円 (x,y,z) \to (x,3xz-y,z), ,円,円 (x,y,z) \to (x,y,3xy-z) $$ and permutations of variables. The sequence of Markoff triples starts with $$ (1,1,1), ,円 (1,1,2), ,円 (1,2,5), ,円 (1,5,13), ,円 (2,5,29), ,円 (1,13,34), ,円 (1,34,89), ,円 (2,29,169), \dots $$ while the sequence of Markoff numbers starts with \begin{equation}\label{eq:Markoffnumbers} 1, 2, 5, 13, 29, 34, 89, 169, \dots \end{equation} Markoff numbers have applications in number theory, group theory and geometry, e.g. they are connected to the Lagrange spectrum arising in Diophantine approximation.

In [BGS26], Bourgain, Gamburd and Sarnak proved the following result.

Theorem 1. Almost all Markoff numbers are composite. Moreover, for every $v\geq 1$, we have $|P_v(N)|=o(|M(N)|)$ as $N\to \infty$, where $M(N)$ in the set of all Markoff numbers up to $N$, and $P_v(N)$ is the set of $m \in M(N)$ that have at most $v$ distinct prime factors.

At the core of the proof is the analysis of Markoff numbers modulo primes $p$. In 1991, Baragar conjectured that that for any prime $p$, and any solution $(a,b,c)\neq (0,0,0)$ to the Markoff equation modulo $p$, there is an integer solution $(x,y,z)$ to the Markoff equation such that $(x,y,z)\equiv (a,b,c),円(\text{mod},円 p)$. Bourgain, Gamburd and Sarnak [BGS26] proved that if $E$ is the set of primes for which this conjecture fails, then $|\{p \in E,円|,円p\leq x\}|=O(x^\epsilon)$ for any $\epsilon>0$, and deduced Theorem 1 from this.

After the arxiv version of [BGS26] appeared, Chen proved Baragar's conjecture with at most finitely many exceptions. Let ${\cal G}_p$ be the graph whose vertices are solutions to Markoff equation modulo $p$, which are connected by edges if and only if they are related by the described-above Vieta jumping operations, again modulo $p$. Chen proved that the number of vertices in any connected component of ${\cal G}_p$ is divisible by $p$. In combination with [BGS26], this implies that ${\cal G}_p$ is in fact connected for all $p>p_0$, where $p_0$ is some unspecified constant. In turn, this implies the truth of Baragar's conjecture for $p>p_0$. Then Eddy et.al. proved that we can take $p_0=3.45\cdot 10^{392}$. On the other hand, in 2025, Brown developed an almost linear time algorithm for testing the Baragar conjecture for any given $p$, and used it to verify the conjecture for all $p<10^6$.

We remark that many other basic questions about Markoff numbers remain open. The most famous one is the unicity conjecture predicting that for any Markoff number $m$ there exists a unique Markoff triple $(x,y,z)$ such that $\max\{x,y,z\}=m$.

[BGS26] Bourgain, J., Gamburd, A. & Sarnak, P. (2026). Strong approximation and Diophantine properties of Markoff triples. Journal of the American Mathematical Society, 39(1), 177-204. https://doi.org/10.1090/jams/1061

• Alexandre Eremenko: Jean Bourgain died in 2018. It is remarkable that his paper is published in 2026.
• bertieb: Vieta jumping operations were demonstrated in a recent Michael Magee Numberphile video, motivated by an equation of the form $x^2+y^2+z^2+w^2=xyzw$ — therefore inasmuch as this non-professional vaguely recognised the context, I would suggest this example is sufficiently elementary

Answer 6

Guth and Maynard proved recently that if $c>17/30$, then the number of primes between $x$ and $x+x^c$ is asymptotically $x^c/\log(x)$. This improves the result of Huxley (1972) which required $c>7/12$.

• David Roberts: That bound is an improvement of 34/60 to 35/60, to avoid others having to make the comparison

Answer 7

I am not sure that this qualifies (doubts about "elementary statement", so I will not object if this post is deleted) but there is a revolutionary result in Computer science which is briefly called $\text{MIP}^*=\text{RE}$. An announcement of its proof is published in a top journal (Communications of the ACM) in 2021. According to the editorial note this paper "is unusual and is still under review". (The complete version is a 223 pages preprint in the arXiv). Since the first announcement in 2020, a mistake has been found but it was corrected by the authors in the same year.

The result (if correct) implies a resolution of several outstanding problems in $C^*$ algebras and quantum mechanics (Connes' Embedding Problem and Tsirelson's Problem).

Apparently people tend to think that the proof is valid since I see papers which use the result.

• Bogdan Grechuk: Thank you. I have already included this in my book based on 2021 publication, which fits to 2001-2025, but if the full version of this paper will be published later, then it fits to this Mathoverflow question as well.
• Glenn Willen: I am able to understand the basic statement of this result, given only an undergraduate background in complexity theory and quantum mechanics, for what it's worth about how elementary it is. Although the exciting implications that are linked here are perhaps beyond me. (Also, wow, this is the first I've heard about it!)
• Alexandre Eremenko: @Glenn Willen: My congratulations!

Answer 8

At the risk of sounding overly egotistical, and that likely some doubt remains among the community about the correctness of my preprint, let me mention my paper Hilbert's tenth problem for systems of diagonal quadratic forms, and Büchi's problem . Before I continue, let me go on the record definitively that V1 on the arxiv, which is 9 pages shorter than subsequent versions, is not correct and has a very significant, likely unfixable gap. V2 introduces a substantially different argument to handle the non-existence of what I call quintuples of Büchi pairs.

The statement of Büchi's Five-Squares Problem (definitely named as such by Leonard Lipshitz in his contribution to Büchi's collected works) is quite simple:

If $x_1, \cdots, x_5$ are positive integers such that $x_1^2, \cdots, x_5^2$ are five increasing integer squares with constant second difference equal to 2ドル$, then $x_{j+1}^2 = (x_1 + j)^2$ for $j = 1,2,3,4$.

Büchi noted that a positive answer to the above implies that Hilbert's Tenth Problem for systems of equations defined by diagonal quadratic forms is undecidable in general. Lipshitz noted that ``the positive existential theory of $\mathbb{N}$ in the language $\mathcal{L} = \{0, 1, +, <, \operatorname{Sq}\}$ is undecidable, where $\operatorname{Sq}(x) \leftrightarrow x$ is a square", provided that the Five-Squares Problem has a positive answer.

In the preprint above, I gave a positive answer to the Five-Squares Problem, thereby obtaining the above two results (stated by Büchi and Lipshitz respectively) as consequences.

I gave a talk on this paper in September at the Number Theory Web Seminar, which is watchable on YouTube: Stanley Yao Xiao: Hilbert's Tenth Problem for systems of diagonal quadratic forms [...] (NTWS 259)

In October, I gave another version of this talk in Saarbrucken.

After I gave the talk in September, Adam Logan soon found a different approach which yields Proposition 7.4 in my paper, which is the key proposition showing the non-existence of non-trivial quintuples of Büchi pairs, by explicitly analyzing the algebraic structure of the underlying scheme. This proof is now written up and included as an appendix in the latest (submitted) version of the paper, which is not available on arxiv but is available upon request.

Currently, there are a few people (myself included) interested in formalizing my proof in Lean, but this is a work in progress.

• D.R.: Wow, I watched your talk. The number of algebraic miracles/"simplifications"/"alignments" that need to happen are just staggering. Syzygy rivaling that of the heavens! At the very edge of human-comprehensible algebraic manipulation. Thanks for sharing :)
• Timothy Chow: On the topic of Hilbert's tenth problem, Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field by Alpöge, Bhargava, Ho, and Shnidman might also qualify. The statement that Hilbert's tenth problem is unsolvable for rings of integers of a number field is understandable to undergraduates.
• mr_e_man: I haven't read either answer carefully, but you seem to contradict math.stackexchange.com/questions/3767695/… . Is there a difference I'm missing?
• Stanley Yao Xiao: @mr_e_man no, there is no contradiction. That question asks for whether it is possible to decide whether a single quadratic equation in arbitrarily many variables is decidable over $\mathbb{Q}$, and this is indeed possible, and solved by Siegel. However, when you restrict to integers and allow arbitrarily many equations (all of which are degree 2ドル$), then it is expected that this is not decidable.
• Stanley Yao Xiao: @SamHopkins I just read the statement again and it is indeed missing the key adjective "non-trivial". Thanks for the catch!

Answer 9

In 1986, Bourgain asked whether any convex body in ${\mathbb R}^n$ of unit volume has a hyperplane section whose $(n-1)$-dimensional volume is at least a universal constant. This statement, known as hyperplane conjecture, or slicing conjecture, turned out to be extremely deep, and later proved to be equivalent to several central questions in probability theory and convex geometry.

One of the equivalent formulations is the manifestation of the fundamental concentration of mass principle, which states that, in high dimensions, most of the mass of an isotropic (see below for the definition of "isotropic") convex body $K$ is concentrated near a sphere with radius $\sqrt{n}$. In the language of the probability theory, this means that if $x$ is selected uniformly at random inside $K$, then $||x||_2 = \sqrt{\langle x,x \rangle}$ is close to $\sqrt{n}$ with high probability.

A uniform distribution in a convex body is a special case of the well-studied family of log-concave distributions. A probability measure $\mu$ on ${\mathbb R}^n$ is called absolutely continuous if there exists a function $g:{\mathbb R}^n \to [0,\infty)$ (called a density) such that $\mu(A)=\int_A g(x)\mathrm{d}x$ for every measurable $A\subset {\mathbb R}^n$. If the set $D=\{x\in{\mathbb R}^n,円|,円g(x)>0\}$ is convex, and $-\log(g(x))$ is a convex function on $D$, then $\mu$ is called log-concave. A density $g$ (and the corresponding probability measure $\mu$) is called isotropic if $$ \int_{{\mathbb R}^n}x ,円g(x),円 \mathrm{d}x = 0 \quad \textrm{and} \quad \int_{{\mathbb R}^n} \langle x,\theta \rangle ,円g(x),円 \mathrm{d}x = 1 \quad \textrm{for every} \quad \theta \in {\mathbb S}^{n-1}. $$

One of the equivalent formulations of the hyperplane conjecture is the inequality $$ {\mathbb P}(||x-y||_2 \leq \epsilon\sqrt{n}) \leq (c_0 \epsilon)^{n}, \quad \forall ,円 y \in {\mathbb R}^n, \quad \forall ,円 \epsilon > 0, $$ where $x$ is an isotropic log-concave vector in ${\mathbb R}^n$, and $c_0>0$ is a universal constant. It states that the probability that $x$ is inside any ball of radius $\epsilon\sqrt{n}$ decreases exponentially with the dimension. In combination with known estimates that $||x||_2$ cannot be much larger than $\sqrt{n}$, this implies that, with high probability, $||x||_2$ is concentrated within a constant multiple of $\sqrt{n}$. Famous thin-shell conjecture (also known as the variance conjecture) is a stronger statement that for any isotropic log-concave vector $x$ in ${\mathbb R}^n$, $$ E[||x||_2 - \sqrt{n}] \leq C $$ for some universal constant $C$, where, as usual, $E$ denotes the expected value. It has long been known that the thin-shell conjecture implies the hyperplane conjecture.

In 2024, Klartag and Lehec, following a large chain of impressive partial results, completed the proof of the hyperplane conjecture.

Theorem 1. The hyperplane conjecture is true.

In 2025, the same authors posted online a preprint, in which they confirmed the thin-shell conjecture as well. In fact, they proved a bit stronger statement.

Theorem 2. Let $x$ be an isotropic, log-concave random vector in $\mathbb{R}^n$. Then, $$ \operatorname{Var}(||x||_2^2) = E\left[ \left(||x||^2_2 - n\right)^2 \right] \leq Cn, $$ where $C > 0$ is a universal constant. In particular, the thin-shell conjecture is true.

The estimate in Theorem 2 is tight, up to the value of the universal constant.

Both theorems are not published in journals yet, hence publication date will be 2026+. These are clearly top-journal-level results, because even partial results towards these conjectures have been published in top journals.

Answer 10

Famous Schur's theorem, published in 1916, states that, for any finite colouring of integers, there must exist same-colour integers $x,y,z$ such that $x+y=z$. In 1974, Hindman proved a far-reaching generalization of this result, stating that, in any finite colouring of integers, there is an infinite set $A$ such that all finite sums $$ S_A=\left\{\left.\sum\nolimits_{x\in B} x,円\right|,円B\subset A, |B|<\infty\right\} $$ of integers from the set $A$ have the same colour.

One of the corollaries of Hindman's work is that, for any finite colouring of integers, there exist infinite sets $B \subset {\mathbb N}$ and $C \subset {\mathbb N}$, such that the set $B+C=\{b+c: b\in B, c\in C\}$ is monochromatic. A conjecture of Erdős, known as the Erdős sumset conjecture predicted that any set $A$ of positive upper density contains a set $B+C$ as above. Nathanson proved in 1980 that this is true if the set $B$ is infinite and $C$ is arbitrarily large but finite. In 2015, Di Nasso, Goldbring, Jin, Leth, Lupini and Mahlburg proved that the set $B+C$ with infinite $B,C$ can be found in any set $A$ of upper density at least 1ドル/2$. Finally, in 2019, Moreira, Richter and Robertson proved the Erdős sumset conjecture in full.

In 2024, Kra, Moreira, Richter and Robertson generalized this result to the sum of any finite number of sets. That is, for any set $A\subset {\mathbb N}$ of positive upper density and every integer $k\geq 1$ there exist infinite sets $B_1,\dots,B_k \subset {\mathbb N}$ with $B_1+\dots+B_k \subset A$, where $$ B_1+\dots+B_k=\{b_1+\dots+b_k: b_1\in B_1, \dots b_k\in B_k\}. $$

In 2025, Hernández, Kousek and Radić generalized this further as follows: for any set $A\subset {\mathbb N}$ of positive upper density, there exists an infinite sequence $B_1, B_2, \dots$ of infinite sets of natural numbers such that $B_1+\dots+B_k \subset A$ for all $k$.

All the above theorems are density versions of the special cases of the original Hindman's theorem. In 1975, Erdős asked to formulate and prove the density version of Hindman's theorem, which is as general as possible. In a paper to be published in 2026, Kra, Moreira, Richter and Robertson achieved this by proving the following theorem.

Theorem 1. For any set $C\subset {\mathbb N}$ of positive upper density and for every integer $k\geq 1$ there is an infinite set $A \subset {\mathbb N}$ and an integer $t\geq 0$ such that $$ S_{A,k}=\left\{\left.\sum\nolimits_{x\in B} x,円\right|,円B\subset A, |B|\leq k\right\} $$ is a subset of $C-t$.

Comparing to Hindman's theorem, Theorem 1 only considers sums with at most $k$ summands, and they belong to a shifted set $C-t$ rather than to $C$ itself. All these restrictions are known to be necessary for the result to be true. Example of odd integers shows that the shift $t$ is necessary, while Straus constructed examples of sets $C$ with density arbitrarily close to 1ドル$ which do not contain a shift of $S_A$ for any infinite $A$.

Answer 11

Chaika and Forni proved that billiard flow for polygonal billiards can be weakly mixing. This solves an important long-open question and will be published in Annals of Mathematics in 2026+.

We consider the motion of an idealized point mass (a "billiard ball") inside the polygon $P$. It follows two rules: (1) When the point is inside $P$, it moves in a straight line at a constant velocity (we usually assume the speed is 1), and (2) When the point hits an edge, it reflects according to the Law of Reflection: the angle of incidence equals the angle of reflection.

To analyze the system mathematically, we define the set of all possible states. A state is determined by the position $x=(x_1,x_2) \in P$ and the direction of motion, represented by a unit vector $v$. The phase space $X$ is the collection of all such pairs $(x,v)$.

The billiard flow describes how the system evolves over time. It is a family of functions $\{\phi_t\}_{t \in \mathbb{R}}$, where $\phi_t$ maps the phase space to itself. If $z_0 \in X$ is the initial state (position and direction), then $\phi_t(z_0)$ is the state of the system after time $t$ has passed, following the rules of billiard motion.

To discuss concepts like "mixing," we need a way to measure the "size" or "volume" of subsets in the phase space and understand how this volume behaves under the flow. A measure $\mu$ is a function that assigns a non-negative number (its size or volume) to subsets of the phase space $X$. For billiards, we use a natural measure called the Liouville measure, which is basically the "uniform" (natural) measure on $X$ normalized such that $\mu(X)=1$.

A flow $\{\phi_t\}$ is measure-preserving if the measure of any measurable set $A \subseteq X$ remains constant as it evolves under the flow. That is, $\mu(\phi_t(A)) = \mu(A)$ for all time $t$. The billiard flow is known to preserve the Liouville measure.

A measure-preserving flow is ergodic if it cannot be decomposed into smaller, independent systems. Formally, if a set $A$ is invariant under the flow (meaning $\phi_t(A) = A$ for all $t$), then $A$ must either have measure zero or full measure ($\mu(A)=0$ or $\mu(A)=1$). The informal intuition is that almost every trajectory will eventually explore the entire phase space.

Mixing concepts describe how thoroughly the flow "stirs" the phase space over time, indicating a tendency towards statistical equilibrium and chaotic behavior. A measure-preserving flow is (strongly) mixing if, for any two measurable sets $A$ and $B$ in the phase space, the probability of starting in $A$ and ending in $B$ after time $t$ approaches the product of their individual probabilities as time goes to infinity. Formally: $$ \lim_{t \to \infty} \mu(\phi_t(A) \cap B) = \mu(A)\mu(B) $$ This indicates that the two events become statistically independent in the limit.

Weak mixing is a condition that is weaker than strong mixing but stronger than ergodicity. It requires that the convergence described in strong mixing happens, but only on average over time. Formally, for any two measurable sets $A$ and $B$: $$ \lim_{T \to \infty} \frac{1}{T} \int_0^T |\mu(\phi_t(A) \cap B) - \mu(A)\mu(B)| dt = 0 $$ The informal intuition is that the system still stirs the phase space and tends towards equilibrium, but it might not do so as rapidly or uniformly as a strongly mixing system. However, the deviations from perfect statistical independence average out to zero over time.

In 1986, Kerckhoff, Masur and Smillie proved the existence of polygons $P$ on which the billiard flow is ergodic. On the other hand, it is widely conjectured that polygonal billiards are never mixing. This naturally leads to the question whether the intermediate property of being weakly mixing is possible for polygonal billiards. In a paper to be published in Annals of Mathematics, Chaika and Forni positively resolved this long-standing open question.

Theorem: For any integer $n\geq 3$, there exists an $n$-vertex polygon $P$ such that the corresponding billiard flow is weakly mixing.