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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.

The lonely runner conjecture is the following open problem:

Conjecture 1 Suppose one has ${n \geq 1}$ runners on the unit circle ${{\bf R}/{\bf Z}}$, all starting at the origin and moving at different speeds. Then for each runner, there is at least one time ${t}$ for which that runner is “lonely” in the sense that it is separated by a distance at least ${1/n}$ from all other runners.

One can normalise the speed of the lonely runner to be zero, at which point the conjecture can be reformulated (after replacing ${n}$ by ${n+1}$) as follows:

Conjecture 2 Let ${v_1,\dots,v_n}$ be non-zero real numbers for some ${n \geq 1}$. Then there exists a real number ${t}$ such that the numbers ${tv_1,\dots,tv_n}$ are all a distance at least ${\frac{1}{n+1}}$ from the integers, thus ${\|tv_1\|_{{\bf R}/{\bf Z}},\dots,\|tv_n\|_{{\bf R}/{\bf Z}} \geq \frac{1}{n+1}}$ where ${\|x\|_{{\bf R}/{\bf Z}}}$ denotes the distance of ${x}$ to the nearest integer.

This conjecture has been proven for ${n \leq 7}$, but remains open for larger ${n}$. The bound ${\frac{1}{n+1}}$ is optimal, as can be seen by looking at the case ${v_i=i}$ and applying the Dirichlet approximation theorem. Note that for each non-zero ${v}$, the set ${\{ t \in {\bf R}: \|vt\|_{{\bf R}/{\bf Z}} \leq r \}}$ has (Banach) density ${2r}$ for any ${0 < r < 1/2}$, and from this and the union bound we can easily find ${t \in {\bf R}}$ for which

$\displaystyle \|tv_1\|_{{\bf R}/{\bf Z}},\dots,\|tv_n\|_{{\bf R}/{\bf Z}} \geq \frac{1}{2n}-\varepsilon$

for any ${\varepsilon>0}$, but it has proven to be quite challenging to remove the factor of ${2}$ to increase ${\frac{1}{2n}}$ to ${\frac{1}{n+1}}$. (As far as I know, even improving ${\frac{1}{2n}}$ to ${\frac{1+c}{2n}}$ for some absolute constant ${c>0}$ and sufficiently large ${n}$ remains open.)

The speeds ${v_1,\dots,v_n}$ in the above conjecture are arbitrary non-zero reals, but it has been known for some time that one can reduce without loss of generality to the case when the ${v_1,\dots,v_n}$ are rationals, or equivalently (by scaling) to the case where they are integers; see e.g. Section 4 of this paper of Bohman, Holzman, and Kleitman.

In this post I would like to remark on a slight refinement of this reduction, in which the speeds ${v_1,\dots,v_n}$ are integers of bounded size, where the bound depends on ${n}$. More precisely:

Proposition 3 In order to prove the lonely runner conjecture, it suffices to do so under the additional assumption that the ${v_1,\dots,v_n}$ are integers of size at most ${n^{Cn^2}}$, where ${C}$ is an (explicitly computable) absolute constant. (More precisely: if this restricted version of the lonely runner conjecture is true for all ${n \leq n_0}$, then the original version of the conjecture is also true for all ${n \leq n_0}$.)

In principle, this proposition allows one to verify the lonely runner conjecture for a given ${n}$ in finite time; however the number of cases to check with this proposition grows faster than exponentially in ${n}$, and so this is unfortunately not a feasible approach to verifying the lonely runner conjecture for more values of ${n}$ than currently known.

One of the key tools needed to prove this proposition is the following additive combinatorics result. Recall that a generalised arithmetic progression (or ${GAP}$) in the reals ${{\bf R}}$ is a set of the form

$\displaystyle P = \{ n_1 v_1 + \dots + n_d v_d: n_1,\dots,n_d \in {\bf Z}; |n_1| \leq N_1, \dots, |n_d| \leq N_d \}$

for some ${v_1,\dots,v_d \in {\bf R}}$ and ${N_1,\dots,N_d > 0}$; the quantity ${d}$ is called the rank of the progression. If ${t>0}$, the progression ${P}$ is said to be ${t}$-proper if the sums ${n_1 v_1 + \dots + n_d v_d}$ with ${|n_i| \leq t N_i}$ for ${i=1,\dots,d}$ are all distinct. We have

Lemma 4 (Progressions lie inside proper progressions) Let ${P}$ be a GAP of rank ${d}$ in the reals, and let ${t \geq 1}$. Then ${P}$ is contained in a ${t}$-proper GAP ${Q}$ of rank at most ${d}$, with

$\displaystyle |Q| \leq (2t)^d d^{6d^2} \prod_{i=1}^d (2N_i+1).$

Proof: See Theorem 2.1 of this paper of Bilu. (Very similar results can also be found in Theorem 3.40 of my book with Van Vu, or Theorem 1.10 of this paper of mine with Van Vu.) $\Box$

Now let ${n \geq 1}$, and assume inductively that the lonely runner conjecture has been proven for all smaller values of ${n}$, as well as for the current value of ${n}$ in the case that ${v_1,\dots,v_n}$ are integers of size at most ${n^{Cn^2}}$ for some sufficiently large ${C}$. We will show that the lonely runner conjecture holds in general for this choice of ${n}$.

let ${v_1,\dots,v_n}$ be non-zero real numbers. Let ${C_0}$ be a large absolute constant to be chosen later. From the above lemma applied to the GAP ${\{ n_1 v_1 + \dots + n_d v_d: n_1,\dots,n_d \in \{-1,0,1\}\}}$, one can find a ${n^{C_0n}}$-proper GAP ${Q}$ of rank at most ${n}$ containing ${\{v_1,\dots,v_n\}}$ such that

$\displaystyle |Q| \leq (6n^{C_0 n})^n n^{6n^2};$

in particular ${|Q| \leq n^{Cn^2}}$ if ${C}$ is large enough depending on ${C_0}$.

We write

$\displaystyle Q = \{ n_1 w_1 + \dots + n_d w_d: n_1,\dots,n_d \in {\bf Z}; |n_1| \leq N_1,\dots,|n_d| \leq N_d \}$

for some ${d \leq n}$, ${w_1,\dots,w_d}$, and ${N_1,\dots,N_d \geq 0}$. We thus have ${v_i = \phi(a_i)}$ for ${i=1,\dots,n}$, where ${\phi: {\bf R}^d \rightarrow {\bf R}}$ is the linear map ${\phi(n_1,\dots,n_d) := n_1 w_1 + \dots + n_d w_d}$ and ${a_1,\dots,a_n \in {\bf Z}^d}$ are non-zero and lie in the box ${\{ (n_1,\dots,n_d) \in {\bf R}^d: |n_1| \leq N_1,\dots,|n_d| \leq N_d \}}$.

We now need an elementary lemma that allows us to create a “collision” between two of the ${a_1,\dots,a_n}$ via a linear projection, without making any of the ${a_i}$ collide with the origin:

Lemma 5 Let ${a_1,\dots,a_n \in {\bf R}^d}$ be non-zero vectors that are not all collinear with the origin. Then, after replacing one or more of the ${a_i}$ with their negatives ${-a_i}$ if necessary, there exists a pair ${a_i,a_j}$ such that ${a_i-a_j \neq 0}$, and such that none of the ${a_1,\dots,a_n}$ is a scalar multiple of ${a_i-a_j}$.

Proof: We may assume that ${d \geq 2}$, since the ${d \leq 1}$ case is vacuous. Applying a generic linear projection to ${{\bf R}^2}$ (which does not affect collinearity, or the property that a given ${a_k}$ is a scalar multiple of ${a_i-a_j}$), we may then reduce to the case ${d=2}$.

By a rotation and relabeling, we may assume that ${a_1}$ lies on the negative ${x}$-axis; by flipping signs as necessary we may then assume that all of the ${a_2,\dots,a_n}$ lie in the closed right half-plane. As the ${a_i}$ are not all collinear with the origin, one of the ${a_i}$ lies off of the ${x}$-axis, by relabeling, we may assume that ${a_2}$ lies off of the ${x}$ axis and makes a minimal angle with the ${x}$-axis. Then the angle of ${a_2-a_1}$ with the ${x}$-axis is non-zero but smaller than any non-zero angle that any of the ${a_i}$ make with this axis, and so none of the ${a_i}$ are a scalar multiple of ${a_2-a_1}$, and the claim follows. $\Box$

We now return to the proof of the proposition. If the ${a_1,\dots,a_n}$ are all collinear with the origin, then ${\phi(a_1),\dots,\phi(a_n)}$ lie in a one-dimensional arithmetic progression ${\{ mv: |m| \leq |Q| \}}$, and then by rescaling we may take the ${v_1,\dots,v_n}$ to be integers of magnitude at most ${|Q| \leq n^{Cn}}$, at which point we are done by hypothesis. Thus, we may assume that the ${a_1,\dots,a_n}$ are not all collinear with the origin, and so by the above lemma and relabeling we may assume that ${a_n-a_1}$ is non-zero, and that none of the ${a_i}$ are scalar multiples of ${a_n-a_1}$.

We write

$\displaystyle a_n-a_1 = (c_1,\dots,c_d) \ \ \ \ \ (1)$

with ${|c_i| \leq 2 N_i}$ for ${i=1,\dots,d}$; by relabeling we may assume without loss of generality that ${c_d}$ is non-zero, and furthermore that

$\displaystyle \frac{|c_i|}{N_i} \leq \frac{|c_d|}{N_d}$

for ${i=1,\dots,d}$. We can also factor

$\displaystyle (c_1,\dots,c_d) = q (c'_1,\dots,c'_d) \ \ \ \ \ (2)$

where ${q}$ is a natural number and ${c'_1,\dots,c'_d}$ have no common factor.

We now define a variant ${\tilde \phi: {\bf R}^d \rightarrow {\bf R}}$ of ${\phi: {\bf R}^d \rightarrow {\bf R}}$ by the map

$\displaystyle \tilde \phi(n_1,\dots,n_d) := n_1 \tilde w_1 + \dots + n_{d-1} \tilde w_{d-1} - \frac{n_d}{c_d} (c_1 \tilde w_1 + \dots + c_{d-1} \tilde w_{d-1}),$

where the ${\tilde w_1,\dots,\tilde w_{d-1}}$ are real numbers that are linearly independent over ${{\bf Q}}$, whose precise value will not be of importance in our argument. This is a linear map with the property that ${\tilde \phi(a_n-a_1)=0}$, so that ${\tilde \phi(a_1),\dots,\tilde \phi(a_n)}$ consists of at most ${n-1}$ distinct real numbers, which are non-zero since none of the ${a_i}$ are scalar multiples of ${a_n-a_1}$, and the ${\tilde w_i}$ are linearly independent over ${{\bf Q}}$. As we are assuming inductively that the lonely runner conjecture holds for ${n-1}$, we conclude (after deleting duplicates) that there exists at least one real number ${\tilde t}$ such that

$\displaystyle \| \tilde t \tilde \phi(a_1) \|_{{\bf R}/{\bf Z}}, \dots, \| \tilde t \tilde \phi(a_n) \|_{{\bf R}/{\bf Z}} \geq \frac{1}{n}.$

We would like to “approximate” ${\phi}$ by ${\tilde \phi}$ to then conclude that there is at least one real number ${t}$ such that

$\displaystyle \| t \phi(a_1) \|_{{\bf R}/{\bf Z}}, \dots, \| t \phi(a_n) \|_{{\bf R}/{\bf Z}} \geq \frac{1}{n+1}.$

It turns out that we can do this by a Fourier-analytic argument taking advantage of the ${n^{C_0 n}}$-proper nature of ${Q}$. Firstly, we see from the Dirichlet approximation theorem that one has

$\displaystyle \| \tilde t \tilde \phi(a_1) \|_{{\bf R}/{\bf Z}}, \dots, \| \tilde t \tilde \phi(a_n) \|_{{\bf R}/{\bf Z}} \leq \frac{1}{10 n^2}$

for a set ${\tilde t}$ of reals of (Banach) density ${\gg n^{-O(n)}}$. Thus, by the triangle inequality, we have

$\displaystyle \| \tilde t \tilde \phi(a_1) \|_{{\bf R}/{\bf Z}}, \dots, \| \tilde t \tilde \phi(a_n) \|_{{\bf R}/{\bf Z}} \geq \frac{1}{n} - \frac{1}{10n^2}$

for a set ${\tilde t}$ of reals of density ${\gg n^{-O(n)}}$.

Applying a smooth Fourier multiplier of Littlewood-Paley type, one can find a trigonometric polynomial

$\displaystyle \eta(x) = \sum_{m: |m| \leq n^{C_0 n/10}} b_m e^{2\pi i mx}$

which takes values in ${[0,1]}$, is ${\gg 1}$ for ${\|x\|_{{\bf R}/{\bf Z}} \geq \frac{1}{n} - \frac{1}{10n^2}}$, and is no larger than ${O( n^{-100 C_0n} )}$ for ${\|x\|_{{\bf R}/{\bf Z}} \leq \frac{1}{n+1}}$. We then have

$\displaystyle \mathop{\bf E}_t \prod_{j=1}^n \eta( t \tilde \phi(a_j) ) \gg n^{-O(n)}$

where ${\mathop{\bf E}_t f(t)}$ denotes the mean value of a quasiperiodic function ${f}$ on the reals ${{\bf R}}$. We expand the left-hand side out as

$\displaystyle \sum_{m_1,\dots,m_n: m_1 \tilde \phi(a_1) + \dots + m_n \tilde \phi(a_n) = 0} b_{m_1} \dots b_{m_n}.$

From the genericity of ${\tilde w_1,\dots,\tilde w_n}$, we see that the constraint

$\displaystyle m_1 \tilde \phi(a_1) + \dots + m_n \tilde \phi(a_n) = 0$

occurs if and only if ${m_1 a_1 + \dots + m_n a_n}$ is a scalar multiple of ${a_n-a_1}$, or equivalently (by (1), (2)) an integer multiple of ${(c'_1,\dots,c'_d)}$. Thus

$\displaystyle \sum_{m_1,\dots,m_n: m_1 a_1 + \dots + m_n a_n \in {\bf Z} (c'_1,\dots,c'_d)} b_{m_1} \dots b_{m_n} \gg n^{-O(n)}. \ \ \ \ \ (3)$

Next, we consider the average

$\displaystyle \mathop{\bf E}_t \varphi( t \xi ) \prod_{j=1}^n \eta( t v_j ) \ \ \ \ \ (4)$

where

$\displaystyle \xi := c'_1 w_1 + \dots + c'_d w_d. \ \ \ \ \ (5)$

and ${\varphi}$ is the Dirichlet series

$\displaystyle \varphi(x) := \sum_{m: |m| \leq n^{C_0 n/2}} e^{2\pi i mx}.$

By Fourier expansion and writing ${v_j = \phi(a_j)}$, we may write (4) as

$\displaystyle \sum_{m,m_1,\dots,m_n: |m| \leq n^{C_0n/2}; m_1 \phi(a_1) + \dots + m_n \phi(a_n) = m \xi} b_{m_1} \dots b_{m_n}.$

The support of the ${b_{m_i}}$ implies that ${|m_i| \leq n^{C_0n/10}}$. Because of the ${n^{C_0 n}}$-properness of ${Q}$, we see (for ${n}$ large enough) that the equation

$\displaystyle m_1 \phi(a_1) + \dots + m_n \phi(a_n) = m \xi \ \ \ \ \ (6)$

implies that

$\displaystyle m_1 a_1 + \dots + m_n a_n \in {\bf Z} (c'_1,\dots,c'_d) \ \ \ \ \ (7)$

and conversely that (7) implies that (6) holds for some ${m}$ with ${|m| \leq n^{C_0 n/2}}$. From (3) we thus have

$\displaystyle \mathop{\bf E}_t \varphi( t \xi ) \prod_{j=1}^n \eta( t v_j ) \gg n^{-O(1)}.$

In particular, there exists a ${t}$ such that

$\displaystyle \varphi( t \xi ) \prod_{j=1}^n \eta( t v_j ) \gg n^{-O(1)}.$

Since ${\varphi}$ is bounded in magnitude by ${n^{C_0n/2}}$, and ${\eta}$ is bounded by ${1}$, we thus have

$\displaystyle \eta(t v_j) \gg n^{-C_0 n/2 - O(1)}$

for each ${1 \leq j \leq n}$, which by the size properties of ${\eta}$ implies that ${\|tv_j\|_{{\bf R}/{\bf Z}} \geq \frac{1}{n+1}}$ for all ${1 \leq j \leq n}$, giving the lonely runner conjecture for ${n}$.