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The self-chosen remit of my blog is “Updates on my research and expository papers, discussion of open problems, and other maths-related topics”.  Of the 774 posts on this blog, I estimate that about 99% of the posts indeed relate to mathematics, mathematicians, or the administration of this mathematical blog, and only about 1% are not related to mathematics or the community of mathematicians in any significant fashion.

This is not one of the 1%.

Mathematical research is clearly an international activity.  But actually a stronger claim is true: mathematical research is a transnational activity, in that the specific nationality of individual members of a research team or research community are (or should be) of no appreciable significance for the purpose of advancing mathematics.  For instance, even during the height of the Cold War, there was no movement in (say) the United States to boycott Soviet mathematicians or theorems, or to only use results from Western literature (though the latter did sometimes happen by default, due to the limited avenues of information exchange between East and West, and former did occasionally occur for political reasons, most notably with the Soviet Union preventing Gregory Margulis from traveling to receive his Fields Medal in 1978 EDIT: and also Sergei Novikov in 1970).    The national origin of even the most fundamental components of mathematics, whether it be the geometry (γεωμετρία) of the ancient Greeks, the algebra (الجبر) of the Islamic world, or the Hindu-Arabic numerals $0,1,\dots,9$, are primarily of historical interest, and have only a negligible impact on the worldwide adoption of these mathematical tools. While it is true that individual mathematicians or research teams sometimes compete with each other to be the first to solve some desired problem, and that a citizen could take pride in the mathematical achievements of researchers from their country, one did not see any significant state-sponsored “space races” in which it was deemed in the national interest that a particular result ought to be proven by “our” mathematicians and not “theirs”.   Mathematical research ability is highly non-fungible, and the value added by foreign students and faculty to a mathematics department cannot be completely replaced by an equivalent amount of domestic students and faculty, no matter how large and well educated the country (though a state can certainly work at the margins to encourage and support more domestic mathematicians).  It is no coincidence that all of the top mathematics department worldwide actively recruit the best mathematicians regardless of national origin, and often retain immigration counsel to assist with situations in which these mathematicians come from a country that is currently politically disfavoured by their own.

Of course, mathematicians cannot ignore the political realities of the modern international order altogether.  Anyone who has organised an international conference or program knows that there will inevitably be visa issues to resolve because the host country makes it particularly difficult for certain nationals to attend the event.  I myself, like many other academics working long-term in the United States, have certainly experienced my own share of immigration bureaucracy, starting with various glitches in the renewal or application of my J-1 and O-1 visas, then to the lengthy vetting process for acquiring permanent residency (or “green card”) status, and finally to becoming naturalised as a US citizen (retaining dual citizenship with Australia).  Nevertheless, while the process could be slow and frustrating, there was at least an order to it.  The rules of the game were complicated, but were known in advance, and did not abruptly change in the middle of playing it (save in truly exceptional situations, such as the days after the September 11 terrorist attacks).  One just had to study the relevant visa regulations (or hire an immigration lawyer to do so), fill out the paperwork and submit to the relevant background checks, and remain in good standing until the application was approved in order to study, work, or participate in a mathematical activity held in another country.  On rare occasion, some senior university administrator may have had to contact a high-ranking government official to approve some particularly complicated application, but for the most part one could work through normal channels in order to ensure for instance that the majority of participants of a conference could actually be physically present at that conference, or that an excellent mathematician hired by unanimous consent by a mathematics department could in fact legally work in that department.

With the recent and highly publicised executive order on immigration, many of these fundamental assumptions have been seriously damaged, if not destroyed altogether.  Even if the order was withdrawn immediately, there is no longer an assurance, even for nationals not initially impacted by that order, that some similar abrupt and major change in the rules for entry to the United States could not occur, for instance for a visitor who has already gone through the lengthy visa application process and background checks, secured the appropriate visa, and is already in flight to the country.  This is already affecting upcoming or ongoing mathematical conferences or programs in the US, with many international speakers (including those from countries not directly affected by the order) now cancelling their visit, either in protest or in concern about their ability to freely enter and leave the country.  Even some conferences outside the US are affected, as some mathematicians currently in the US with a valid visa or even permanent residency are uncertain if they could ever return back to their place of work if they left the country to attend a meeting.  In the slightly longer term, it is likely that the ability of elite US institutions to attract the best students and faculty will be seriously impacted.  Again, the losses would be strongest regarding candidates that were nationals of the countries affected by the current executive order, but I fear that many other mathematicians from other countries would now be much more concerned about entering and living in the US than they would have previously.

It is still possible for this sort of long-term damage to the mathematical community (both within the US and abroad) to be reversed or at least contained, but at present there is a real risk of the damage becoming permanent.  To prevent this, it seems insufficient for me for the current order to be rescinded, as desirable as that would be; some further legislative or judicial action would be needed to begin restoring enough trust in the stability of the US immigration and visa system that the international travel that is so necessary to modern mathematical research becomes “just” a bureaucratic headache again.

Of course, the impact of this executive order is far, far broader than just its effect on mathematicians and mathematical research.  But there are countless other venues on the internet and elsewhere to discuss these other aspects (or politics in general).  (For instance, discussion of the qualifications, or lack thereof, of the current US president can be carried out at this previous post.) I would therefore like to open this post to readers to discuss the effects or potential effects of this order on the mathematical community; I particularly encourage mathematicians who have been personally affected by this order to share their experiences.  As per the rules of the blog, I request that “the discussions are kept constructive, polite, and at least tangentially relevant to the topic at hand”.

I’ve just uploaded to the arXiv my paper “Some remarks on the lonely runner conjecture“, submitted to Contributions to discrete mathematics. I had blogged about the lonely runner conjecture in this previous blog post, and I returned to the problem recently to see if I could obtain anything further. The results obtained were more modest than I had hoped, but they did at least seem to indicate a potential strategy to make further progress on the problem, and also highlight some of the difficulties of the problem.

One can rephrase the lonely runner conjecture as the following covering problem. Given any integer “velocity” ${v}$ and radius ${0 < \delta < 1/2}$, define the Bohr set ${B(v,\delta)}$ to be the subset of the unit circle ${{\bf R}/{\bf Z}}$ given by the formula

$\displaystyle B(v,\delta) := \{ t \in {\bf R}/{\bf Z}: \|vt\| \leq \delta \},$

where ${\|x\|}$ denotes the distance of ${x}$ to the nearest integer. Thus, for ${v}$ positive, ${B(v,\delta)}$ is simply the union of the ${v}$ intervals ${[\frac{a-\delta}{v}, \frac{a+\delta}{v}]}$ for ${a=0,\dots,v-1}$, projected onto the unit circle ${{\bf R}/{\bf Z}}$; in the language of the usual formulation of the lonely runner conjecture, ${B(v,\delta)}$ represents those times in which a runner moving at speed ${v}$ returns to within ${\delta}$ of his or her starting position. For any non-zero integers ${v_1,\dots,v_n}$, let ${\delta(v_1,\dots,v_n)}$ be the smallest radius ${\delta}$ such that the ${n}$ Bohr sets ${B(v_1,\delta),\dots,B(v_n,\delta)}$ cover the unit circle:

$\displaystyle {\bf R}/{\bf Z} = \bigcup_{i=1}^n B(v_i,\delta). \ \ \ \ \ (1)$

Then define ${\delta_n}$ to be the smallest value of ${\delta(v_1,\dots,v_n)}$, as ${v_1,\dots,v_n}$ ranges over tuples of distinct non-zero integers. The Dirichlet approximation theorem quickly gives that

$\displaystyle \delta(1,\dots,n) = \frac{1}{n+1}$

and hence

$\displaystyle \delta_n \leq \frac{1}{n+1}$

for any ${n \geq 1}$. The lonely runner conjecture is equivalent to the assertion that this bound is in fact optimal:

Conjecture 1 (Lonely runner conjecture) For any ${n \geq 1}$, one has ${\delta_n = \frac{1}{n+1}}$.

This conjecture is currently known for ${n \leq 6}$ (see this paper of Barajas and Serra), but remains open for higher ${n}$.

It is natural to try to attack the problem by establishing lower bounds on the quantity ${\delta_n}$. We have the following “trivial” bound, that gets within a factor of two of the conjecture:

Proposition 2 (Trivial bound) For any ${n \geq 1}$, one has ${\delta_n \geq \frac{1}{2n}}$.

Proof: It is not difficult to see that for any non-zero velocity ${v}$ and any ${0 < \delta < 1/2}$, the Bohr set ${B(v,\delta)}$ has Lebesgue measure ${m(B(v,\delta)) = 2\delta}$. In particular, by the union bound

$\displaystyle m(\bigcup_{i=1}^n B(v_i,\delta)) \leq \sum_{i=1}^n m(B(v_i,\delta)) \ \ \ \ \ (2)$

we see that the covering (1) is only possible if ${1 \leq 2 n \delta}$, giving the claim. $\Box$

So, in some sense, all the difficulty is coming from the need to improve upon the trivial union bound (2) by a factor of two.

Despite the crudeness of the union bound (2), it has proven surprisingly hard to make substantial improvements on the trivial bound ${\delta_n \geq \frac{1}{2n}}$. In 1994, Chen obtained the slight improvement

$\displaystyle \delta_n \geq \frac{1}{2n - 1 + \frac{1}{2n-3}}$

which was improved a little by Chen and Cusick in 1999 to

$\displaystyle \delta_n \geq \frac{1}{2n-3}$

when ${2n-3}$ was prime. In a recent paper of Perarnau and Serra, the bound

$\displaystyle \delta_n \geq \frac{1}{2n-2+o(1)}$

was obtained for arbitrary ${n}$. These bounds only improve upon the trivial bound by a multiplicative factor of ${1+O(1/n)}$. Heuristically, one reason for this is as follows. The union bound (2) would of course be sharp if the Bohr sets ${B(v_i,\delta)}$ were all disjoint. Strictly speaking, such disjointness is not possible, because all the Bohr sets ${B(v_i,\delta)}$ have to contain the origin as an interior point. However, it is possible to come up with a large number of Bohr sets ${B(v_i,\delta)}$ which are almost disjoint. For instance, suppose that we had velocities ${v_1,\dots,v_s}$ that were all prime numbers between ${n/4}$ and ${n/2}$, and that ${\delta}$ was equal to ${\delta_n}$ (and in particular was between ${1/2n}$ and ${1/(n+1)}$. Then each set ${B(v_i,\delta)}$ can be split into a “kernel” interval ${[-\frac{\delta}{v_i}, \frac{\delta}{v_i}]}$, together with the “petal” intervals ${\bigcup_{a=1}^{v_i-1} [\frac{a-\delta}{v_i}, \frac{a+\delta}{v_i}]}$. Roughly speaking, as the prime ${v_i}$ varies, the kernel interval stays more or less fixed, but the petal intervals range over disjoint sets, and from this it is not difficult to show that

$\displaystyle m(\bigcup_{i=1}^s B(v_i,\delta)) = (1-O(\frac{1}{n})) \sum_{i=1}^s m(B(v_i,\delta)),$

so that the union bound is within a multiplicative factor of ${1+O(\frac{1}{n})}$ of the truth in this case.

This does not imply that ${\delta_n}$ is within a multiplicative factor of ${1+O(1/n)}$ of ${\frac{1}{2n}}$, though, because there are not enough primes between ${n/4}$ and ${n/2}$ to assign to ${n}$ distinct velocities; indeed, by the prime number theorem, there are only about ${\frac{n}{4\log n}}$ such velocities that could be assigned to a prime. So, while the union bound could be close to tight for up to ${\asymp n/\log n}$ Bohr sets, the above counterexamples don’t exclude improvements to the union bound for larger collections of Bohr sets. Following this train of thought, I was able to obtain a logarithmic improvement to previous lower bounds:

Theorem 3 For sufficiently large ${n}$, one has ${\delta_n \geq \frac{1}{2n} + \frac{c \log n}{n^2 (\log\log n)^2}}$ for some absolute constant ${c>0}$.

The factors of ${\log\log n}$ in the denominator are for technical reasons and might perhaps be removable by a more careful argument. However it seems difficult to adapt the methods to improve the ${\log n}$ in the numerator, basically because of the obstruction provided by the near-counterexample discussed above.

Roughly speaking, the idea of the proof of this theorem is as follows. If we have the covering (1) for ${\delta}$ very close to ${1/2n}$, then the multiplicity function ${\sum_{i=1}^n 1_{B(v_i,\delta)}}$ will then be mostly equal to ${1}$, but occasionally be larger than ${1}$. On the other hand, one can compute that the ${L^2}$ norm of this multiplicity function is significantly larger than ${1}$ (in fact it is at least ${(3/2-o(1))^{1/2}}$). Because of this, the ${L^3}$ norm must be very large, which means that the triple intersections ${B(v_i,\delta) \cap B(v_j,\delta) \cap B(v_k,\delta)}$ must be quite large for many triples ${(i,j,k)}$. Using some basic Fourier analysis and additive combinatorics, one can deduce from this that the velocities ${v_1,\dots,v_n}$ must have a large structured component, in the sense that there exists an arithmetic progression of length ${\asymp n}$ that contains ${\asymp n}$ of these velocities. For simplicity let us take the arithmetic progression to be ${\{1,\dots,n\}}$, thus ${\asymp n}$ of the velocities ${v_1,\dots,v_n}$ lie in ${\{1,\dots,n\}}$. In particular, from the prime number theorem, most of these velocities will not be prime, and will in fact likely have a “medium-sized” prime factor (in the precise form of the argument, “medium-sized” is defined to be “between ${\log^{10} n}$ and ${n^{1/10}}$“). Using these medium-sized prime factors, one can show that many of the ${B(v_i,\delta)}$ will have quite a large overlap with many of the other ${B(v_j,\delta)}$, and this can be used after some elementary arguments to obtain a more noticeable improvement on the union bound (2) than was obtained previously.

A modification of the above argument also allows for the improved estimate

$\displaystyle \delta(v_1,\dots,v_n) \geq \frac{1+c-o(1)}{2n} \ \ \ \ \ (3)$

if one knows that all of the velocities ${v_1,\dots,v_n}$ are of size ${O(n)}$.

In my previous blog post, I showed that in order to prove the lonely runner conjecture, it suffices to do so under the additional assumption that all of the velocities ${v_1,\dots,v_n}$ are of size ${O(n^{O(n^2)})}$; I reproduce this argument (slightly cleaned up for publication) in the current preprint. There is unfortunately a huge gap between ${O(n)}$ and ${O(n^{O(n^2)})}$, so the above bound (3) does not immediately give any new bounds for ${\delta_n}$. However, one could perhaps try to start attacking the lonely runner conjecture by increasing the range ${O(n)}$ for which one has good results, and by decreasing the range ${O(n^{O(n^2)})}$ that one can reduce to. For instance, in the current preprint I give an elementary argument (using a certain amount of case-checking) that shows that the lonely runner bound

$\displaystyle \delta(v_1,\dots,v_n) \geq \frac{1}{n+1} \ \ \ \ \ (4)$

holds if all the velocities ${v_1,\dots,v_n}$ are assumed to lie between ${1}$ and ${1.2 n}$. This upper threshold of ${1.2 n}$ is only a tiny improvement over the trivial threshold of ${n}$, but it seems to be an interesting sub-problem of the lonely runner conjecture to increase this threshold further. One key target would be to get up to ${2n}$, as there are actually a number of ${n}$-tuples ${(v_1,\dots,v_n)}$ in this range for which (4) holds with equality. The Dirichlet approximation theorem of course gives the tuple ${(1,2,\dots,n)}$, but there is also the double ${(2,4,\dots,2n)}$ of this tuple, and furthermore there is an additional construction of Goddyn and Wong that gives some further examples such as ${(1,2,3,4,5,7,12)}$, or more generally one can start with the standard tuple ${(1,\dots,n)}$ and accelerate one of the velocities ${v}$ to ${2v}$; this turns out to work as long as ${v}$ shares a common factor with every integer between ${n-v+1}$ and ${2n-2v+1}$. There are a few more examples of this type in the paper of Goddyn and Wong, but all of them can be placed in an arithmetic progression of length ${O(n \log n)}$ at most, so if one were very optimistic, one could perhaps envision a strategy in which the upper bound of ${O(n^{O(n^2)})}$ mentioned earlier was reduced all the way to something like ${O( n \log n )}$, and then a separate argument deployed to treat this remaining case, perhaps isolating the constructions of Goddyn and Wong (and possible variants thereof) as the only extreme cases.