Distribution of boundary points of expansion and application to the lonely runner conjecture
aa r X i v : . [ m a t h . C O ] S e p DISTRIBUTION OF BOUNDARY POINTS OF EXPANSION ANDAPPLICATION TO THE LONELY RUNNER CONJECTURE
T. AGAMA
Abstract.
In this paper we study the distribution of boundary points of ex-pansion. As an application, we say something about the lonely runner problem.We show that given k runners S i round a unit circular track with the conditionthat at some time ||S i − S i +1 || = ||S i +1 − S i +2 || for all i = 1 , . . . , k −
2, thenat that time we have ||S i +1 − S i || > D ( n ) πk − i = 1 , . . . , k − D ( n ) > n . In particular, we show that given atmost eight S i ( i = 1 , , . . . ,
8) runners running round a unit circular track withdistinct constant speed and the additional condition ||S i − S i +1 || = ||S i +1 −S i +2 || for all 1 ≤ i ≤ s >
1, then at that time their mutualdistance must satisfy the lower bound ||S i − S i +1 || > π C √ C > ≤ i ≤ Introduction
The lonely runner conjecture is the assertion that given n runners round a unitcircle with constant distinct speed and starting at a common time and place, theremust exist a time for which their mutual distances should be at least n . Theconjecture has been verified for many special cases. For instance in [2], it has beenshown that the conjecture hold for six runners. It is also shown in [1] for at most seven runners. In this paper, by studying the distribution of boundary points of anexpansion, we verify this conjecture in it’s crude form with an extra conditioningfor at most eight runners. We obtain a conditional result of this conjecture byshowing that: Theorem 1.1.
Given k runners S i round a unit circular track with the conditionthat at some time ||S i − S i +1 || = ||S i +1 − S i +2 || for all i = 1 , . . . , k − , then atthat time we have ||S i +1 − S i || > D ( n ) πk − for all i = 1 , . . . , k − and where D ( n ) > is a constant depending on the degreeof a certain polynomial of degree n . In particular, we show that
Date : September 6, 2019.2000
Mathematics Subject Classification.
Primary 54C40, 14E20; Secondary 46E25, 20C20.
Key words and phrases.
Lonely runner; boundary points.
Theorem 1.2.
Let S i ( i = 1 , , . . . , ) be runners running round the unit circulartrack. Under the condition ||S i − S i +1 || = ||S i +1 − S i +2 || for all ≤ i ≤ at sometime s > , then ||S i − S i +1 || > π D √ for some constant D > for all ≤ i ≤ . Definitions and background
Definition 2.1.
Let S = ( f , f , . . . , f n ) such that each f i ∈ R [ x ]. By the derivativeof S denoted ∇ ( S ), we mean ∇ ( S ) = (cid:18) df dx , df dx , . . . , df n dx (cid:19) . We denote the derivative of this tuple at a point a ∈ R to be ∇ a ( S ) = (cid:18) df ( a ) dx , df ( a ) dx , . . . , df n ( a ) dx (cid:19) . Definition 2.2.
Let S = ( f , f , . . . , f n ) such that each f i ∈ R [ x ]. By the integralof S denoted ∆( S ), we mean∆( S ) = (cid:18) Z f ( x ) dx, . . . , Z f n ( x ) dx (cid:19) . The corresponding integral between the points S a = ( a , . . . , a n ) and S b = ( b , . . . , b n ),denoted ∆ S a , S b ( S ) is given by∆ S a , S b ( S ) = (cid:18) b Z a f ( x ) dx, . . . , b n Z a n f n ( x ) dx (cid:19) . Definition 2.3.
Let {S i } ∞ i =1 be a collection of tuples of R [ x ]. Then by an expansionon {S i } ∞ i =1 , we mean the composite map γ − ◦ β ◦ γ ◦ ∇ : {S i } ∞ i =1 −→ {S i } ∞ i =1 , where γ ( S ) = f f ... f n and β ( γ ( S )) = · · ·
11 0 · · · · · · ...1 1 · · · f f ... f n . Definition 2.4.
Let {S j } ∞ j =1 be a collection of tuples of R [ x ]. By the boundarypoints of the n th expansion, denoted Z [( γ − ◦ β ◦ γ ◦ ∇ ) n ( S j )], we mean the set Z [( γ − ◦ β ◦ γ ◦ ∇ ) n ( S j )] := (cid:8) ( a , a , . . . , a n ) : Id i [( γ − ◦ β ◦ γ ◦ ∇ ) na i ( S j )] = 0 (cid:9) . ISTRIBUTION OF BOUNDARY POINTS OF EXPANSION AND APPLICATION TO THE LONELY RUNNER CONJECTURE3 Distribution of boundary points of expansion
In this section we study the distribution of the boundary points of any phase ofexpansion. We first introduce the notion of integration of polynomials along theboundaries of various phases of expansion, which we then use as a main tool. Welaunch the following definition in that regard.
Definition 3.1.
Let f ( x ) = c n x n + c n − x n − + · · · + c x + c be a polynomial ofdegree n , then we call the tuple S f = ( c n x n , c n − x n − , . . . , c x + c )= ( g ( x ) , g ( x ) , . . . , g n ( x ))the tuple representation of f . By the integral of f ( x ) along the boundary of the m th phase expansion, we mean the formal integral Z B m ( S f ) m 1) is the unit tuple, and −−−−−−−→ O ∆ S i , S i +1 and −−→ O S e are theposition vectors of ∆ S i , S i +1 and S e , respectively, with S i = ( a , a , . . . , a n ) and S i +1 = ( b , b , . . . , b n ). Remark . It is in practice very difficult to ascertain the local distribution ofboundary points of expansion. However, we can show that if we shrink the spacebounded by the boundary of an expansion, then points on the boundary should beclosely packed in some sense. We use the notion of integration along boundaries asa black box. Theorem 3.3. Let f ( x ) := c n x n + c n − x n − + · · · + c x + c be a polynomial ofdegree n . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B m ( S f ) m Proof. Let f ( x ) = c n x n + c n − x n − + · · · + c x + c ∈ R [ x ] be a polynomial ofdegree n and suppose (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B m ( S f ) m 0. By a repeated application of the triangle inequality, we find that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B m ( S f ) m 0, it follows that there exist some S i , S i +1 ∈ Z [( γ − ◦ β ◦ γ ◦∇ ) m ( S f )] with ||S i − S i +1 || < ||S i − S j || for all j = i + 1. It follows that for someclosest pair of boundary points, the inequality ǫ ( B m ( S f ) − M √ n < p | a − b | + · · · + | a n − b n | is valid, and thus it must be that ||S i − S i +1 || > δ by choosing δ = ǫ ( B m ( S f ) − M √ n . Conversely, suppose there exist some closest boundary point S i , S i +1 ∈ Z [( γ − ◦ β ◦ γ ◦ ∇ ) m ( S f )] such that ||S i − S i +1 || > δ for some δ := δ ( n ) > 0. Then it follows that p | a − b | + · · · + | a n − b n | > δ .By choosing R = min {| g i ( x ) | : x ∈ [ a i , b i ] } ni =1 , we find that ||−−−−−−−−−−→ O ∆ S i , S i +1 ( S f ) || = vuuut | b Z a g ( x ) dx | + · · · + | b n Z a n g n ( x ) dx | ≥ R p | a − b | + · · · + | a n − b n | = δR. ISTRIBUTION OF BOUNDARY POINTS OF EXPANSION AND APPLICATION TO THE LONELY RUNNER CONJECTURE5 It follows that B m ( S f ) − X i =1 X S i , S i +1 ∈B m ( S f ) ||S i || < ||S i +1 || −−−−−−−−−−→ O ∆ S i , S i +1 ( S f ) · −−→ O S e > B m ( S f ) − X i =1 X S i , S i +1 ∈B m ( S f ) ||S i || < ||S i +1 || δR ||−−→ O S e || cos α = δ ( B m ( S f ) − R √ n cos α where α is the angle between the vectors −−−−−−−−−−→ O ∆ S i , S i +1 ( S f ) and −−→ O S e . It follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B m ( S f ) m 0. The result follows by taking δ := ǫ ( B m ( S f ) − CR √ n | cos α | . (cid:3) Remark . Theorem 3.3 in the affirmative tells us that we can use the area as ayardstick to determine the distribution of points on the boundary of any phase ofexpansion. 4. Rotation of the boundary of expansion In this section we introduce the concept of rotation of the boundary of an ex-pansion. Definition 4.1. Let ( γ − ◦ β ◦ γ ◦ ∇ ) m ( S j ) be an expansion with correspondingboundary B m ( S j ). Then we say the map ∨ is a rotation of the boundary B m ( S j ) if ∨ : B m ( S j ) −→ B m ( S j ) . We say an expansion admits a rotation if there exist such a map. In other words,we say the map ∨ induces a rotation on the expansion. We say the boundary isstable under rotation if || ∨ ( S a ) || ≈ ||S a || for S a ∈ B m ( S j ). Otherwise we say it isunstable. Remark . Next we prove a result that indicates that boundary points of anexpansion whose boundary occupies a small enough region must be stable. Proposition 4.1. Let f ( x ) := c n x n + · · · + c x + c ∈ R [ x ] , a polynomial with degree n ≥ . Let ( γ − ◦ β ◦ γ ◦ ∇ ) m ( S f ) be an expansion with corresponding boundary B m ( S f ) admits a rotation ∨ . If (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B m ( S f ) m Proof. Let f ( x ) := c n x n + · · · + c x + c ∈ R [ x ], a polynomial with degree n ≥ γ − ◦ β ◦ γ ◦ ∇ ) m ( S f ) be an expansion with corresponding boundary B m ( S f )admits a rotation ∨ . Suppose also that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B m ( S f ) m Let ( γ − ◦ β ◦ γ ◦ ∇ ) m ( S j ) be an expansion with correspondingboundary B m ( S j ). Then we say the map ∨ is a rotation of the boundary B m ( S j )with frequency s if ∨ s : B m ( S j ) −→ B m ( S j ) , where ∨ s = ∨ ◦ ∨ ◦ · · · ◦ ∨ is the s -fold rotation on the boundary of expansion. Remark . It is important to recognize that rotation with frequency s is the timefor which points on the boundary of expansion are allowed to be in motion by aninduced rotation. Proposition 4.2. Let ( γ − ◦ β ◦ γ ◦ ∇ ) m ( S j ) be an expansion with correspondingboundary B m ( S j ) . Then any permutation σ : B m ( S j ) −→ B m ( S j ) where σ ( S i ) = S σ ( i ) for ≤ i ≤ B m ( S j ) for S i ∈ B m ( S j ) is a rotation of theboundary of expansion. Spherical defoliation of the boundary of expansion Definition 5.1. Let B m ( S f ) and S k − be the boundary of the m th expansion andthe k dimensional unit sphere, respectively. Then by the spherical defoliation ofthe boundary of expansion, we mean the mapΛ : B m ( S f ) −→ S k − such that for any S a ∈ B m ( S f ), then we haveΛ( S a ) = S a ||S a || . ISTRIBUTION OF BOUNDARY POINTS OF EXPANSION AND APPLICATION TO THE LONELY RUNNER CONJECTURE7 Application to the Lonely runner conjecture Theorem 6.1. Given k runners S i round a unit circular track with the conditionthat at some time ||S i − S i +1 || = ||S i +1 − S i +2 || for all i = 1 , . . . , k − , then atthat time we have ||S i +1 − S i || > D ( n ) πk − for all i = 1 , . . . , k − and where D ( n ) > is a constant depending on the degreeof a certain polynomial of degree n .Proof. First assume any polynomial g ( x ) := b x n + · · · + b x + b for some choiceof n so that the size of the boundary of expansion B m ( S g ) = k for some m < n .The under the condition ||S i − S i +1 || = ||S i +1 − S i +2 || for all i = 1 , . . . , k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B m ( S g ) g ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = π and apply the s -fold rotation ∨ s = ∨ ◦ ∨ ◦ · · · ◦ ∨ on the boundary B m ( S g ). Thenby Proposition 4.1 points on this boundary are now unstable for time s > ||S i +1 − S i || > C ( n ) πk − i = 1 , . . . , k − D ( n ) > n . Since some point on the boundary of expansion maynot be a point on the unit circle, we apply the spherical defoliation Λ : B m ( S g ) −→ S k , and we obtain ||S i +1 − S i || > D ( n ) πk − i = 1 , . . . , k − D ( n ) > n . (cid:3) Lemma 6.2. Let f ( x ) := c x + c x + c x + c be a polynomial of degree andsuppose ||S i − S i +1 || = ||S i +1 − S i +2 || for all ≤ i ≤ . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B ( S f ) f ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = π if and only if ||S i − S i +1 || > π C √ for some constant C > for all ≤ i ≤ .Proof. The result follows by taking n = 3 in Theorem 3.3. (cid:3) T. AGAMA Theorem 6.3. Let S i ( i = 1 , , . . . , ) be runners running round the unit circulartrack. Under the condition ||S i − S i +1 || = ||S i +1 − S i +2 || for all ≤ i ≤ at sometime s > then ||S i − S i +1 || > π D √ for some constant D > for all ≤ i ≤ .Proof. Let f ( x ) := c x + c x + c x + c be a polynomial of degree 3 and suppose ||S i − S i +1 || = ||S i +1 − S i +2 || for all 1 ≤ i ≤ 6. Then we set (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B ( S f ) f ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = π and apply a rotation ∨ s with frequency s > B ( S f ). Then byProposition 4.1, boundary points of expansion are now unstable for time s > ||S i − S i +1 || > π C √ C > ≤ i ≤ 7. By applying the defoliation Λ : B ( S f ) −→ S ,we obtain ||S k − S k +1 || > π D √ D > ≤ k ≤ (cid:3) . References 1. Barajas, Javier and Serra, Oriol, The lonely runner with seven runners , the electronic journalof combinatorics, vol.15:1 , 2008, pp 48.2. Bohman, Tom and Holzman, Ron and Kleitman, Dan, Six lonely runners , the electronicjournal of combinatorics, vol. 8:2, Oldenbourg wissenschaftsverlag, 2001, pp 3. Department of Mathematics, African Institute for Mathematical science, Ghana E-mail address : [email protected]/[email protected]