THE LONELY RUNNER PROBLEM FOR MANY RUNNERS. Artūras Dubickas Vilnius University, Lithuania

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1 GLASNIK MATEMATIČKI Vol. 46(66)(2011), THE LONELY RUNNER PROBLEM FOR MANY RUNNERS Artūras Dubickas Vilius Uiversity, Lithuaia Abstract. The loely ruer cojecture asserts that for ay positive iteger ad ay positive umbers v 1 < < v there exists a positive umber t such that v i t 1/(+1) for every i = 1,...,. We verify this cojecture for uder assumptio that the speeds of the ruers satisfy log for j = 1,..., Itroductio Let be a positive iteger, ad let v 1 < v 2 < < v be positive real umbers. The Loely Ruer Cojecture asserts that there is a positive umber t such that (1.1) v i t 1 +1 for every i = 1,2,...,. Throughout y stads for the distace betwee a real umber y ad the earest iteger to y. Note that iequality (1.1) is optimal if, for istace, v i = vi for each i = 1,...,, where v > 0 is a fixed real umber (see, e.g., [6]). For some there are also other values of v i s whe equality i (1.1) is attaied (see [12]). The cojecture origially comes from the paper of Wills ([17]), where it is stated for iteger v i s. Idepedetly, this problem was cosidered by Cusick ([7]). The ame of the loely ruer cojecture comes from the followig beautiful iterpretatio of the problem due to Goddy ([5]). Suppose k ruers havig distict costat speeds start at a commo poit ad ru laps o a circular track with circumferece 1. The for ay give ruer there is a time at which that ruer is at least 1/k (alog the track) away from every 2010 Mathematics Subject Classificatio. 11J13, 11J25, 11J71. Key words ad phrases. Loely ruer cojecture, Diophatie approximatio, Lovász local lemma. 25

2 26 A. DUBICKAS other ruer. Takig k = +1 ad assumig that the speeds of the ruers are u 0 < u 1 < < u, we see that at the time t > 0 the ruer with speed, say, u 0 is at distace 1/(+1) from all other ruers if ad oly if (1.1) holds for v i = u i u 0, i = 1,...,. It seems that the loely ruer cojecture is very deep i geeral. It is kow that it has some useful applicatios to so-called view-obstructio problems, flows i regular matroids, chromatic umbers for distace graphs, etc. ([5,7,8,13]). The problem has bee settled for = 2,3 ([4]), = 4 (first i [8] ad the a simpler proof was foud i [5]), = 5 ([6], a simpler proof i [15]). Recetly, Barajas ad Serra ([2]) proved the cojecture for = 6. For each 7 the loely ruer cojecture is still ope. O the other had, there aresome coditios o the speeds ofthe ruers v 1 < < v uder which the loely ruer cojecture holds. If, for example, (1.2) v /v 1 the takig t 0 = 1/(+1)v 1 it is easy to see that the umbers v i t 0 = v i /(+ 1)v 1, i = 1,...,, all lie i the iterval [1/(+1),/(+1)], so (1.1) holds. Recetly, Padey ([14]) showed that the coditio (1.3) +1 2(+1) 1 for each j = 1,..., 1 implies (1.1). This iequality (i a slightly differet form) was also obtaied by Ruzsa, Tuza ad Voigt ([16]), ad the the costat 2(+1)/( 1) was improved to 2 i [3]. Usig the same method of ested itervals as i [14] oe ca easily prove (1.1) uder coditio (1.4) for j = 1,..., 1 which is slightly weaker tha (1.3) (see the begiig of Sectio 2). I this ote we prove the followig: Theorem 1.1. Let 32, ad let v 1 < v 2 < < v be positive real umbers satisfyig +[(+1)/12e] (+1) for each j = 1,2,..., [(+1)/12e]. The there is a positive umber t such that v i t > 1/(+1) for each i = 1,2,...,. Here ad below, [y] stads for the itegral part of a real umber y. Theorem 1.1 implies the followig improvemet of the coditios (1.3), (1.4) uder which (1.1) holds: Corollary 1.2. Suppose that κ is a costat strictly greater tha 8e = The there is a positive iteger (κ) such that for each iteger

3 THE LONELY RUNNER PROBLEM FOR MANY RUNNERS 27 (κ) ad each collectio of positive umbers v 1 < v 2 < < v satisfyig +1 (1.5) 1+ κlog for every j = 1,2,..., 1 there is a positive umber t such that (1.6) v i t > 1/(+1) for every i = 1,2,...,. I particular, for κ = 33, oe ca take (33) = Note that the coditio log, j = 1,..., 1, of Corollary 1.2 with κ = 22 > 8e yields v /v 1 (1 + 22log ) 1. Here, the right had side is approximately 22 for large. Comparig with (1.2) we see that there is still a polyomial gap betwee ad 22 for the bouds o v /v 1 for which the loely ruer cojecture is ot verified. At least this gap is smaller tha a correspodig expoetial gap betwee ad (roughly) 2 1 which comes from (1.3) ad (1.4). We remark that, by Lemma 6 i [10], for ay positive umbers v 1 < < v ad ay ε > 0 ad T > 0 there is a iterval I = [u 0,u 0 +ε/2v ], where u 0 > T, such that v i t < ε for each t I ad each i = 1,...,. This shows that all the ruers ca be arbitrarily close to their startig positio at arbitrarily large time t. The referee poited out that this problem is somewhat related to Bogolyubov s theorem o Bohr eighborhoods ad, despite some similarity to the loely ruer, has a differet ature. We shall derive Theorem 1.1 from Lemma 2.1 below. Sice the proof of the lemma is based o a so-called Lovász local lemma (see [1] ad [11]), the Lovász lemma is implicitly preset i the proofs below. 2. Proofs We first prove that (1.4) implies (1.1). Ideed, settig I 1 := [1/( + 1)v 1,/(+1)v 1 ] we see that v 1 t 1/(+1) for each t I 1. Put k 1 := 0. We claim that there is a sequece of ested itervals I 1 I of the form I i := [(k i +1/(+1))/v i,(k i +/(+1))/v i ] with iteger k i for i = 1,...,. The v i t 1/( + 1) for each i = 1,...,. The proof is by iductio. Assume that we have such sequece of ested itervals I 1 I j, where 1 j 1. Note that the iterval [ vj+1 k j + +1/ 1, +1k j + (+1/ 1) ]

4 28 A. DUBICKAS cotais a positive iteger, say, k j+1, because the legth of this iterval is 1, by (1.4). From +1 k j + +1/ 1 +1 k j+1 +1k j + (+1/ 1) +1 we deduce that I j+1 := [(k j+1 +1/(+1))/+1,(k j+1 +/(+1))/+1 ] I j. This completes the iductio step ad so proves that (1.4) implies (1.1). The ext lemma is Theorem 1.1 with dimesio m = 1 from [9]. Lemma 2.1. Let (ξ k ) k=1 be a sequece of real umbers. If h is a positive iteger, c(h) is a real umber greater tha 4eh ad (t k ) k=1 is a sequece of positive umbers satisfyig t k+h c(h)t k for each iteger k 1 the there is a real umber x such that t k x ξ k > 1 2c(h) for every k 1. Take ξ k := 0 for each k 1. Put t k := v k for k = 1,..., ad, say, t k := v c(h) k for k +1. By Lemma 2.1, there is a real umber x such that (2.1) v k x > 1 2c(h) for k = 1,...,. Sice y = y,the same iequality holds for t := x > 0 istead of x. Obviously, x 0, so t > 0. Put h := [( + 1)/12e] ad c(h) := + 1. Note that h 1, because 32. Sice e / Q, we have h < ( +1)/12e. Thus the right had side of (2.1) is 1 2c(h) = 1 8e[(+1)/12e] 1 2(+1) > 12e 8e(+1) 1 2(+1) = Therefore, the iequality v i t > 1/( + 1) holds for i = 1,..., provided that v i+h ( + 1)v i for i = 1,..., h. This is exactly the coditio of Theorem 1.1. The proof of the theorem is completed. We first prove that oe ca take (33) = i Corollary 1.2. Assume that iequality (1.5) holds with κ = 33. To apply Theorem 1.1 we will check with Maple that ( 1+ 33log ) h ( = 1+ 33log ) [(+1)/12e] > +1 for each iteger Ideed, the fuctio [ z +1 ] ( g(z) := log 1+ 33logz ) log(z +1) 12e z is positive for z except for two itervals J 1 ad J 2 such that J 1 (16373,16374) ad J 2 (16406,16407). At the poits z =

5 THE LONELY RUNNER PROBLEM FOR MANY RUNNERS ,16374,16406,16407 the fuctio g(z) is positive. Thus g() > 0 for each iteger For the proof of Corollary 1.2 we assume that (1.5) holds with some κ > 8e. We shall derive iequality (1.6) directly from Lemma 2.1. Set ǫ := (k 8e)/(4e+1). The ǫ > 0 satisfies (2.2) 8e(1+ǫ/2) = κ ǫ. This time, we select h := [(+1)/(κ ǫ)] ad c(h) := [(+1)/ǫ]+1. The, by (2.2), the right had side of (2.1) is 1 2c(h) = > 1 8e[(+1)/(κ ǫ)] 1 2([(+1)/ǫ]+1) κ ǫ 8e(+1) ǫ 2(+1) = Hece, by Lemma 2.1, iequality (1.6) holds for every i = 1,..., ad some t > 0 provided that v i+h ([(+1)/ǫ]+1)v i for each i = 1,..., h. Note that (1.5) implies v i+h (1+ κlog ) h v i for i = 1,..., h. Sice h 1 for κ, it remais to prove the iequality ( (2.3) 1+ κlog ) [(+1)/(κ ǫ)] [(+1)/ǫ]+1 for each sufficietly large. It is clear that κ/(κ ǫ) > 1+ǫ/κ. Thus there is a positive iteger 1 = 1 (ǫ,κ) = 1 (κ) such that the left had side of (2.3) is greatertha 1+ǫ/κ for 1. O the other had, there is a positive iteger 2 = 2 (ǫ,κ) = 2 (κ) such that the right had side of (2.3) is at most 2/ε < 1+ǫ/κ for 2. Thus (2.3) holds for each max( 1 (κ), 2 (κ)). This completes the proof of the corollary. Ackowledgemets. I thak the referee who quickly wrote a detailed report cotaiig some useful iformatio. Refereces [1] N. Alo ad J. Specer, The probabilistic method, Wiley-Itersciece, New York, 2000, 2d ed. [2] J. Barajas ad O. Serra, The loely ruer with seve ruers, Electro. J. Combi. 15 (2008), R48, 18 pp. [3] J. Barajas ad O. Serra, O the chromatic umber of circulat graphs, Discrete Math. 309 (2009), [4] U. Betke ad J.M. Wills, Utere Schrake für zwei diophatische Approximatios- Fuktioe, Moatsh. Math. 76 (1972), [5] W. Bieia, L. Goddy, P. Gvozdjak, A. Sebő ad M. Tarsi, Flows, view obstructios ad the loely ruer, J. Combi. Theory Ser. B 72 (1998), 1 9. [6] T. Bohma, R. Holzma ad D. Kleitma, Six loely ruers, Electro. J. Combi. 8(2) (2001), R3, 49 pp.

6 30 A. DUBICKAS [7] T. W. Cusick, View-obstructio problems i -dimesioal geometry, J. Combi. Theory Ser. A 16 (1974), [8] T. W. Cusick ad C. Pomerace, View-obstructio problems III, J. Number Theory 19 (1984), [9] A. Dubickas, A approximatio by lacuary sequece of vectors, Combi. Probab. Comput. 17 (2008), [10] A. Dubickas, A approximatio property of lacuary sequeces, Israel J. Math. 170 (2009), [11] P. Erdős ad L. Lovász, Problems ad results of 3-chromatic hypergraphs ad some related questios, i: A. Hajal et al., eds. Ifiite ad fiite sets (dedic. to P. Erdős o his 60th birthday), Vol. II. North-Hollad, Amsterdam, 1975, pp [12] L. Goddy ad E. B. Wog, Tight istaces of the loely ruer, Itegers 6 (2006), #A38, 14 pp. [13] D. D. F. Liu, From raibow to the loely ruer: a survey o colorig parameters of distace graphs, Taiwaese J. Math. 12 (2008), [14] R. K. Padey, A ote o the loely ruer cojecture, J. Iteger Seq. 12 (2009), Article , 4 pp. [15] J. Reault, View-obstructio: a shorter proof for 6 loely ruers, Discrete Math. 287 (2004), [16] I. Ruzsa, Zs. Tuza ad M. Voigt, Distace graphs with fiite chromatic umber, J. Combi. Theory Ser. B 85 (2002), [17] J. M. Wills, Zwei Sätze über ihomogee diophatische Approximatio vo Irratioalzahle, Moatsh. Math. 71 (1967), A. Dubickas Departmet of Mathematics ad Iformatics Vilius Uiversity Naugarduko 24, Vilius LT Lithuaia arturas.dubickas@mif.vu.lt Received: Revised: