Correlation among runners and some results on the Lonely Runner Conjecture

The Lonely Runner Conjecture was posed independently by Wills and Cusick and has many applications in different mathematical fields, such as diophantine approximation. This well-known conjecture states that for any set of runners running along the unit circle with constant different speeds and starting at the same point, there is a moment where all of them are far enough from the origin. We study the correlation among the time that runners spend close to the origin. By means of these correlations, we improve a result of Chen on the gap of loneliness and we extend an invisible runner result of Czerwinski and Grytczuk. In the last part, we introduce dynamic interval graphs to deal with a weak version of the conjecture thus providing some new results.


Introduction
The Lonely Runner Conjecture was posed independently by Wills [21] in 1967 and Cusick [10] in 1982. Its picturesque name comes from the following interpretation due to Goddyn [5]. Consider a set of k runners on the unit circle running with different constant speeds and starting at the origin. The conjecture states that, for each runner, there is a time where she is at distance at least 1/k on the circle from all the other runners.
For any real number x, denote by x , the distance from x to the closest integer By assuming that one of the runners has zero speed, the conjecture can be easily seen to be equivalent to the following one.

G. Perarnau and O. Serra
Conjecture 1 (Lonely Runner Conjecture). For every n ≥ 1 and every set of nonzero speeds v 1 , . . . , v n , there exists a time t ∈ R such that tv i ≥ 1 n + 1 , If true, the Lonely Runner Conjecture is best possible: for the set of speeds, there is no time for which all the runners are further away from the origin than 1 n+1 . An infinite family of additional extremal sets for the conjecture can be found in [17].
The conjecture is obviously true for n = 1, since at some point tv 1 = 1/2, and it is also easy to show that it holds for n = 2. Many proofs for n = 3 are given in the context of diophantine approximation (see [4,10]).
A computer-assisted proof for n = 4 was given by Cusick and Pomerance motivated by a view-obstruction problem in geometry [11], and later Biena et al. [5] provided a simpler proof by connecting it to nowhere zero flows in regular matroids. The conjecture was proved for n = 5 by Bohmann, Holzmann and Kleitman [6]. Barajas and Serra [3] showed that the conjecture holds for n = 6 by studying the regular chromatic number of distance graphs.
In [6], the authors also showed that the conjecture can be reduced to the case where all speeds are positive integers and in the sequel we will assume this to be the case. In particular, we also may assume that t takes values on the (0, 1) unit interval, since if t ∈ Z, tv i = 0 for all i ∈ [n].
Czerwiński [12] proved a strengthening of the conjecture if all the speeds are chosen uniformly at random among all the n-subsets of [N ]. In particular, Czerwiński's result implies that, for almost all sets of runners, as N → ∞ there is a time where all the runners are arbitrarily close to 1/2. The dependence of N with respect to n for which this result is valid was improved by Alon [1] in the context of colorings of Cayley graphs.
Dubickas [14] used a result of Peres and Schlag [20] in lacunary integer sequences to prove that the conjecture holds if the sequence of increasing speeds grows fast enough; in particular, for n sufficiently large, if for every 1 ≤ i < n. These results introduce the use of the Lovász Local Lemma to deal with the dependencies among the runners.
Another approach to the conjecture is to reduce the gap of loneliness. That is, to show that, for some fixed δ ≤ 1 n+1 and every set of nonzero speeds, there exists a time t ∈ (0, 1) such that For this approach it is particularly useful to define the following sets, For every t ∈ A i , we will say that the i-th runner is δ-close to the origin at time t. Otherwise, we will say that the runner is δ-far from the origin at time t.
The set A i can be thought of as an event in the probability space (0, 1) with the uniform distribution. Notice that we have Pr(A i ) = 2δ independently from the value of v i . In this setting, if then, there exists a time t for which (3) holds.
Here, it is also convenient to consider the indicator random variables X i for the events A i . Let X = n i=1 X i count the number of runners which are δ-close from the origin at a time t ∈ (0, 1) chosen uniformly at random.
A first straightforward result in this direction is obtained by using the union bound in (4). For any δ < 1 2n , we have This result was improved by Chen [7] who showed that, for every set of n nonzero speeds, there exists a time for every i ∈ [n].
If 2n − 3 is a prime number, then the previous result was extended by Chen and Cusick [8]. In this case, these authors proved that, for every set of n speeds, there exists a time t ∈ R such that for every i ∈ [n].
In order to improve (5), we exactly compute the pairwise join probabilities Pr(A i ∩ A j ), the amount of time that two runners spend close to the origin at the same time. As a corollary, we give the following lower bound on E(X 2 ).
Then, we are able to improve Chen's result on the gap of loneliness around the origin.
Theorem 3. For every sufficiently large n and every set v 1 , . . . , v n of nonzero speeds there exists a time t ∈ (0, 1) such that , The proof of Theorem 3 uses a Bonferroni-type inequality due to Hunter [19] (see Lemma 14) that improves the union bound with the knowledge of pairwise intersections.
The bound on δ in Theorem 3 can be substantially improved in the case of sets of speeds taken from a sequence with divergent sum of inverses. More precisely the following result is proven.
Theorem 4. For every set v 1 , . . . , v n of nonzero speeds there exists a time t ∈ (0, 1) such that The condition (2) of Dubickas [14] implies that the conjecture is true if the set of speeds grows as a polynomial of high degree. Theorem 4 is interesting in the sense that it provides meaningful bounds for the opposite case, that is when the speeds grow slowly. In particular, if , . This is a natural density condition on the set of speeds and covers the cases where the speeds are strongly correlated.
Another interesting result on the Lonely Runner Conjecture, was given by Czerwiński and Grytczuk [13]. We say that a runner k is almost alone at time t if there exists a j = k such that for every i = j, k. If this case we say that j leaves k almost alone.
In [13], the authors showed that every runner is almost alone at some time. This means that Conjecture 1 is true if we are allowed to make one runner invisible, that is, there exists a time when all runners but one are far enough from the origin.
Theorem 5 ( [13]). For every n ≥ 1 and every set of nonzero speeds v 1 , . . . , v n , there exist a time t ∈ (0, 1) such that the origin is almost alone at time t.
As a corollary of Proposition 2, we get the following result that extends Theorem 5 when n is large.
This theorem extends Theorem 5 by showing the existence of not only one but two runners whose deletion leave the origin alone at some point.
A similar result can be derived by using a model of dynamic circular interval graphs. By using it we can show that at least two runners are almost alone at the same time.
This paper is organized as follows. In Section 2 we compute the pairwise join probabilities for the events A i and give a proof for Proposition 2. As a corollary of these results, we also prove Theorem 3 and 4 (Subsection 2.1), and Theorem 6 (Subsection 2.2). In Section 3 we introduce an approach on the problem based on dynamic circular interval graphs and prove Theorem 7. Finally, in Section 4 we give some conclusions and discuss some open questions.

Correlation among runners
In this section we want to study the pairwise join probabilities Pr that, if A i and A j were independent events, then we would have Pr . This is not true in the general case, but, as we will see later on, some of these pairwise probabilities can be showed to be large enough.
where gcd(v i , v j ) denotes the greatest common divisor of v i and v j and {·} is the fractional part.
Let us also consider the function f : The proofs of Propositions 8 and 9 below are based on the proofs of Lemmas 3.4 and 3.5 in Alon and Ruzsa [2].
Let us start by studying the case when the speeds v i and v j are coprime.

Proposition 8.
Let v i and v j be coprime positive integers and 0 < δ < 1. Then Proof. Observe that A i and A j can be expressed as the disjoint unions of intervals Denote by I = (−α, α) and J = (−β, β). We have where in the last equality we used the fact that gcd(v i , v j ) = 1.
For each −1/2 < x < 1/2, define d(x) = Pr(I ∩ (J + x)). Let us assume that v j < v i . We can write d(x) as follows (see Figure 1): By symmetry, we have Write αv i v j = p + ε ji and βv i v j = q + ε ij , where p and q are integers.
Observe that and that Then, Thus, Proposition 8 can be easily generalized to pairs of speeds that are not coprime.
Let v i and v j be positive integers and 0 < δ < 1. Then The proof follows by applying Proposition 8 to v i and v j , which are coprime.
Corollary 10. For every v i and v j we have Proof. We observe that, for x, y ≤ 1, we have that min(x, y) ≥ xy and thus f (x, y) ≥ −xy. Therefore Proposition 9 leads to the following lower bound, By using (8), we can provide a first lower bound on the second moment of X, We devote the rest of this section to improve (10). Let us first show for which values is f nonnegative.
The following lemma shows that the error term of Pr(A i ∩ A j ) provided in Proposition 9, cannot be too negative if v i and v j are either close or far enough from each other.
Proof. For the sake of simplicity, let us write l being nonnegative integers and 0 ≤ x, y < δ −1 . In particular, observe that ε ij = xδ and ε ji = yδ. Moreover, we can assume that v i and v j are such that f (ε ij , ε ji ) is negative, otherwise, there is nothing to prove.
We split the proof in the two different cases each consisting of some other subcases. Figure 2 illustrates the subcase considered in each situation.
where the last inequality holds from the fact that f (ε ij , ε ji ) < 0 and y ≤ v j / gcd(v i , v j ).

G. Perarnau and O. Serra
This expression is minimized in the same point as in subcase B.3, x = γk 2−γ δ −1 and y = (1 − γ)x − (γk − 1)δ −1 . Hence, we have for any k ≥ 0. The following lemma shows that among a large set of positive numbers, there should be a pair of numbers satisfying that they are either close or far enough from each other.
Lemma 13. For every c > 1, T > c and every set x 1 ≥ · · · ≥ x m+1 > 0 of nonnegative numbers, with m ≥ log c T , there is a pair i, j ∈ [m + 1] such that Proof. Suppose that for each pair i < j we have x i > cx j . In particular, for each i ≤ m, we have x i > cx i+1 and x 1 > c m x m+1 ≥ T x m+1 . Hence the second possibility holds for i = 1 and j = m + 1.
For any fixed ε > 0, we call a pair i, j ∈ [n] ε-good if Pr(A i ∩ A j ) ≥ (1 − ε)4δ 2 . Now we are able to improve the lower bound on the second moment of X given in (10).
Proof of Proposition 2. Recall that by (8), for any pair i, j ∈ [n], we have Pr(A i ∩ A j ) ≥ 2δ 2 . We will show that at least a Ω 1 log δ −1 fraction of the pairs are ε-good.  (1)) c ε n 2 2 log δ −1 , ε-good unordered pairs. Now, we are able to give a lower bound on the second moment, for some c ε that depends only on ε.
Next, we show some applications of our bounds that extend some known results.

First application: Improving the gap of loneliness
In this subsection we show how to use the result of Proposition 9 on the pairwise join probabilities to prove Theorem 3. To this end we will use the following Bonferroni-type inequality due to Hunter [19] (see also Galambos and Simonelli [16]) that slightly improves the union bound in the case where the events are not pairwise disjoint.
Lemma 14 (Hunter [19]). For any tree T with vertex set V (T ) = [n], we have As we have already mentioned, Pr(A i ) = 2δ. Thus, it remains to select a tree T that maximizes ij∈E(T ) Pr(A i ∩ A j ).
Proof. Recall that Proposition 9 states that Set M to be the largest integer satisfying M = γ −1 < (2ε ) −1 . We will construct a large forest F on the set of vertices [n], where all the edges ij ∈ E(F ) are ε -good. In particular they will satisfy, Let us show how to select the edges of the forest by a procedure. Set S 0 = [n] and E 0 = ∅. In the k-th step, If no such pair exists, we stop the procedure.
Let τ be the number of steps that the procedure runs before being halted. By Lemma 13 with c = (1 − γ) −1 and T = (γδ) −1 we can always find such an edge ij, provided that the set S k has size at least log c (αδ −1 ). Thus τ ≥ n − log c T . Since the size of the sets E k increases exactly by one at each step, we have (1))n. Besides, by construction E τ is an acyclic set of edges: since we delete one of the endpoints of each selected edge from the set S k , E τ induces a 1-degenerate graph or equivalently, a forest.
By Lemma 12, for each edge ij in E τ we have Therefore we can construct a spanning tree T on the vertex set [n], that contains the forest F and thus satisfies if n is large enough.
Let us proceed to prove Theorem 3.
Theorem 4 follows from the following Corollary.
Corollary 16. For every sufficiently large n and every set of nonzero speeds v 1 , . . . , v n , such that there exists a tree T , then there exists a time t ∈ (0, 1) such that for every i ∈ [n]. In particular, if n i=2 1 vi = c the same conclusion follows.
Proof. By using inequality (9) we have Using Lemma 11 in a similar way as in the proof of Theorem 3, we obtain that Pr The last part of the corollary follows by considering T to be the star with center in the smallest of the speeds.

Second Application: Invisible runners
This subsection is devoted to the proof of Theorem 6. In particular, we use the result of Proposition 2 to show that there is a large fraction of time where only one runner is δ-close to the origin, for δ = 1 n+1 . This implies the existence of at least two runners whose deletion leave the origin alone at some time.
Proof of Theorem 6. For δ = 1 n+1 , we have E(X) = 2n n+1 = 2 − 2 n+1 . Moreover, by Proposition 2, there exists some small constant c > 0 such that, For every 0 ≤ k ≤ n, let p k = Pr(X = k) the probability that exactly k runners are δ-close to the origin. We may assume that p 0 = 0 since otherwise, there would exists a time when all the runners are δ-far from the origin, which already implies Conjecture 1. Then we have the following system of linear equations, From these equations one can deduce that, Then, p 1 is minimized when p 3 = · · · = p n−1 = 0 and p n = (1 + O(n −1 )) 2c (n−1)(n−2) log n . Thus, Since a runner spends no more than 2δ = 2 n+1 fraction of the time δ-close the origin and c does not depend on n, there should be at least two such runners that leave the origin almost alone at some time.

Weaker conjectures and interval graphs
In this section we give a proof for Theorem 7. The following weaker conjecture has been proposed by Spencer 1 .
Conjecture 17 (Weak Lonely Runner Conjecture). For every n ≥ 1 and every set of different speeds v 1 , . . . , v n , there exist a time t ∈ R and a runner j ∈ [n], such that For every set S ⊆ [n], we say that S is isolated at time t if, Observe that S = {i} is isolated at time t, if and only v i is lonely at time t.
To study the appearance of isolated sets, it is convenient to define a dynamic graph G(t), whose connected components are sets of isolated runners at time t. For each 1 ≤ i ≤ n and t ∈ (0, 1), define the following dynamic interval in (0, 1) associated to the i-th runner, In other words, I i (t) is the interval that starts at the position of the i-th runner at time t and has length 1 n .
Now we can define the following dynamic circular interval graph G(t) = (V (t), E(t)). The vertex set V (t) is composed by n vertices u i that correspond to the set of runners, and two vertices u i and u j are connected if and only if I i (t) ∩ I j (t) = ∅ (see Figure 3).
Observation 18. The graph G(t) satisfies the following properties, 2. Each connected component of G(t), correspond to an isolated set of runners at time t.
3. If u i is isolated in G(t), then v i is alone at time t. 4. All the intervals have the same size, |I i (t)| = 1/n, and thus, G(t) is a unit circular interval graph.
We can restate the Lonely Runner Conjecture in terms of the dynamic interval graph G(t).
Conjecture 19 (Lonely Runner Conjecture). For every i ≤ n there exists a time t such that u i is isolated in Let µ be the uniform measure in the unit circle. For every subgraph H ⊆ G(t) we define µ(H) = µ(∪ ui∈V (H) I i (t)), the length of the arc occupied by the intervals corresponding to H. Notice that, if H contains an edge, then since the intervals I i (t) are closed in one extreme but open in the other one. If H consists of isolated vertices, then (14) does not hold.
The dynamic interval graph G(t) allows us to prove a very weak version of the conjecture. Let us assume that v 1 < v 2 < · · · < v n .
Proposition 20. There exist a time t ∈ R and a nonempty subset S ⊂ [n] such that S is isolated at time t.
Proof. Let t be the minimum number for which the equation tv n − 1 = tv 1 − 1/n holds. This is the first time that the slowest runner v 1 is at distance exactly 1/n ahead from the fastest runner v n .
For the sake of contradiction, assume that there is just one connected component of order n. Note that u 1 u n / ∈ E(G(t)) and since G(t) is connected, there exists a path in G(t) connecting u 1 and u n . By (14), we have µ(G) < 1. Thus, there is a point x ∈ (0, 1) such that x / ∈ I i (t) for every i ∈ [n].
Observe that, if one of the parts in Proposition 20 consists of a singleton, say S = {i}, then Conjecture 17 would be true.
Let us show how to use the dynamic graph to prove an invisible lonely runner theorem, similar to Theorem 5.
Proposition 21. There exists t ∈ (0, 1) such that G(t) has either some isolated vertex or it has at least two vertices of degree one.
Proof. Define Y : (0, 1) → N by, Let t ∈ (0, 1) be chosen uniformly at random. Then Y (t) is a random variable over 0, 1, . . . , n 2 . We will show that E(Y (t)) ≤ (n − 1). If we are able to do so, by a first moment argument, we know that there exists a time t 0 for which Y (t 0 ) ≤ n − 1. Then, denoting by d i the degree of u i , we have In the dynamic interval graph setting, an invisible runner is equivalent to a vertex u with a neighbor of degree one, say v. If u is removed, then v becomes isolated in G(t) and thus, alone in the runner setting. Thus, Theorem 7 is a direct corollary of Proposition 21.

Concluding remarks and open questions
1. In Proposition 2 we gave a lower bound for E(X 2 ). We believe that this proof can be adapted to show that E(X 2 ) is larger.
The proof of this conjecture relies on showing that either most pairs are ε-good or the contribution of the positive error terms is larger than the contribution of the negative ones. On the other hand, notice that it is not true that E(X 2 ) is O(δ 2 n 2 ). For the set of speeds in (1), Cilleruelo [9] has shown that E(X 2 ) = (1 + o(1)) 12 π 2 δn log n , which is a logarithmic factor away from the lower bound in Proposition 2, when δ = Θ(n −1 ). It is an open question whether (15) also holds as an upper bound for E(X 2 ).
2. Ideally, we would like to estimate the probabilities Pr(∩ i∈S A i ), for every set S ⊆ [n] of size k. In general, it is not easy to compute such probability. As in (15), the sum of the k-wise join probabilities are not always of the form O(δ k n k ). However it seems reasonable to think that, for every set S, we have where c k depends only on k. Moreover, we know that c k ≤ 2 k , since this is the case when the speeds {v i } i∈S are rationally independent and the conjecture holds (see e.g. Horvat and Stoffregen [18].) The inequality 8 shows that c 2 = 2. In general, we also conjecture that