A generalization of the 3d distance theorem
aa r X i v : . [ m a t h . N T ] F e b A GENERALIZATION OF THE 3D DISTANCE THEOREM
MANISH MISHRA AND AMY BINNY PHILIP
Abstract.
Let P be a positive rational number. Call a function f : R → R to have finite gaps property mod P if the following holds: for any positiveirrational α and positive integer M , when the values of f ( mα ), 1 ≤ m ≤ M ,are inserted mod P into the interval [0 , P ) and arranged in increasing order,the number of distinct gaps between successive terms is bounded by a constant k f which depends only on f . In this note, we prove a generalization of the 3ddistance theorem of Chung and Graham. As a consequence, we show that apiecewise linear map with rational slopes and having only finitely many non-differentiable points has finite gaps property mod P . We also show that if f isdistance to the nearest integer function, then it has finite gaps property mod1 with k f ≤ introduction The well known three gaps theorem was first observed by H. Steinhaus andproved independently by V. T. S´os [6,7] and others [8,9] (see [1] for a nice summaryand recent generalization). The three gaps theorem is a special case ( d = 1) of thefollowing more general theorem of Chung and Graham [2]. Theorem (3d distance theorem) . Let α > be an irrational number and N , . . . , N d be positive integers. When the fractional parts of d arithmetic sequences nα + k i , k i ∈ R , n = 1 , . . . , N i , ≤ i ≤ d are inserted into a circle of unit circumference,the gaps between successive terms takes at most d distinct values. What makes this theorem surprising is the fact that the fractional parts of thesequence nα are known to be uniformly distributed in the interval [0 , λ , let I λ be the discrete set { nλ | n ∈ Z } . For x ∈ R ,define λ -floor ⌊ x ⌋ λ and λ -roof ⌈ x ⌉ λ as: ⌊ x ⌋ λ = max { r ∈ I λ | r ≤ x } , ⌈ x ⌉ λ = min { r ∈ I λ | r ≥ x } . Define λ -fractional part functions {−} ′ λ and {−} ′′ λ as: { x } ′ λ = ( x − ⌊ x ⌋ λ if x ≥ x − ⌈ x ⌉ λ if x < , { x } ′′ λ = x − ⌊ x ⌋ λ . Define { x } ′∞ = x . We write { x } ′′ as { x } .Choose a λ -fractional part function {−} ′ λ or {−} ′′ λ and denote it by {−} λ . Weprove the following generalization of the 3d distance theorem. Theorem 1.1.
Let α > be an irrational number, N , . . . , N d be positive integersand n , . . . , n d be non-negative integers such that n i ≤ N i , ≤ i ≤ d . Write N = P di =1 ( N i − n i ) . Consider the linear maps ˆ f i : x ∈ R p i q x + k i ∈ R , ≤ i ≤ d ,where = p i ∈ Z , q ∈ Z > and k i ∈ R . Fix P to be a positive rational and let λ be any positive integer multiple of P q . Define f i : R → R by f i ( x ) = ˆ f i ( { x } λ ) . Insertmod P , the values of f i ( mα ) , n i < m ≤ N i , ≤ i ≤ d , in the interval [0 , P ) toform an increasing sequence ( b n ) ≤ n ≤ N . Write ℓ = lcm( p , . . . , p d ) > , c i := ℓ/p i and c = P di =1 | c i | . Then there are at most c distinct values in the set of gaps g m defined by g = P + b − b N , g m = b m − b m − , m = 2 , . . . , N. Note that Theorem 1.1 allows the possibility of some points to coincide. Theordering of coincidental points is defined in Section 2.Let || · || : R → [0 , /
2] denote the distance to the nearest integer function. Bydefinition || x || = min( {| x |} , − {| x |} ) . As a special case of Theorem 1.1, we obtain the following result which was provedin [3] using different methods.
Corollary 1.2.
Let α > be an irrational number and M > be an integer. Whenthe values || nα || , ≤ n ≤ M , are arranged in ascending order in the interval [0 , ] ,the gaps between successive terms may take at most distinct values. Our proof of Corollary 1.2 is significantly shorter than the proof in [3]. However,the bound obtained in loc. cit. is effective.
Corollary 1.3.
Let f : R → R be a piecewise linear map with rational slopesand having only finitely many non-differentiable points. Let α > be an irrationalnumber and M > be an integer. For any positive rational P , when the values f ( mα ) , ≤ m ≤ M , are inserted mod P in [0 , P ) and arranged in ascending order,the gaps between successive terms may take at most k f distinct values, where k f isa constant which depends only on f . Our proof of Theorem 1.1 is an adaptation of the elegant proof of the 3 d distanceTheorem by Liang [4]. 2. Proof of Theorem 1.1
Proof.
For 1 ≤ i ≤ d , let B i be the set of all triples β im = ( γ im , i, m ) ∈ [0 , P ) × Z × Z where γ im ∼ = f i ( mα ) mod P , n i ≤ m ≤ N i . Write B = S di =1 B i . We give a strictordering ≺ on B by declaring β im ≺ β jn , iff γ im < γ jn or γ im = γ jn and [ i < j or ( i = j and m < n )] . Arrange the elements of B in a strictly increasing sequence ( b n ) ≤ n ≤ N with thisordering. Applying arithmetic modulo P , we identify P with 0 and consider { γ im | n i ≤ m ≤ N i , ≤ i ≤ d } as living in this circle [0 , P ]. This makes the ordering on B a cyclic ordering, which we again denote by ≺ . Thus b N and b are consecutivein this cyclic ordering. To simplify notation, will often abuse notation and write β im when we mean γ im .A gap interval is an interval in the circle [0 , P ] of the form [ β in , β jm ] where β in , β jm are consecutive points of B in the cyclic ordering ≺ . Write ℓ = ℓ/q . A gapinterval is rigid if translating a gap interval by ℓ α does not produce a gap interval.Observe that gap intervals cannot loop upon successive translations by ℓ α . To seethis, suppose s is a positive integer such that translation by sℓ α maps [ β in , β jm ]to itself. Then either β in + sℓ α = β in and β jm + sℓ α = β jm , or β in + sℓ α = β jm and β jm + sℓ α = β in . Either of these cases contradicts the irrationality of α . GENERALIZATION OF THE 3D DISTANCE THEOREM 3
Now, a gap interval I is rigid if upon translation by ℓ α it produces an interval J for which one of the following holds:(i) At least one of the end points of J is not in B .(ii) The translated interval J has endpoints in B but they are not consecutive.For case (i), let β in be an end point of I such that β in + ℓ α = β i ( n + c i ) / ∈ B .Then in particular, β i ( n + c i ) / ∈ B i . If c i >
0, then β i ( n + c i ) / ∈ B i iff n + c i > N i .Then, β in will be in the set S i = { β im | N i − c i + 1 ≤ m ≤ N i } . If c i <
0, then β i ( n + c i ) / ∈ B i iff n + c i ≤ n i . Then β in will be in the set T i = { β im | n i ≤ m ≤ − c i + n i } . Call the elements of S i and T i to be starting points . Then for each i , the startingpoints have cardinality | c i | . Since each starting point is the boundary of at mosttwo gap intervals, case (i) contributes at most 2 P di =1 | c i | = 2 c rigid intervals.For case (ii), let β kp ∈ B be an internal point of J . Then β k ( p − c k ) is an internalpoint of I . Since I is a gap interval, this implies that β k ( p − c k ) / ∈ B . In particular β k ( p − c k ) / ∈ B k . This implies that β kp belongs to the set T ′ k = { β km | n k ≤ m ≤ c k + n k } or S ′ k = { β km | N k + c k + 1 ≤ m ≤ N k } according as c k > c k <
0. Call the elements of S ′ i and T ′ i to be finish points .Then for each i , the finish points have cardinality at most | c i | . Thus case (ii)contributes at most P di =1 | c k | = c rigid intervals.We have shown that there can be at most 3 c distinct rigid intervals and conse-quently at most 3 c gap interval sizes. This completes the proof. (cid:3) Proof of Corollaries 1.2 and 1.3
Proof of Corollary 1.2.
We retain the notations of Theorem 1.1 and its proof inSection 2. Put d = 2, q = 1 p = 1, p = − k = 0 , k = 1, N = N = M , P = 1 = λ and {−} λ = {−} ′′ λ . Then c = 2, the starting points are { M α } and1 − { α } , and the finish points are 1 − { M α } and { α } . Now write D ⊂ B for theset of points {|| mα || | ≤ m ≤ N } . Since α is irrational, the points of B are alldistinct. Arrange the points in D in usual increasing order. Since the ordering ≺ on B is the usual order on the circle [0 , B \ D ⊂ ( , u, v are consecutive points of D , then it they are also consecutive points of B .Consequently, it follows from Theorem 1.1 that the number of distinct gap valuesin D is at most 3 c = 6. (cid:3) Remark 3.1.
When α is a positive cube root of 15, we get four distinct gap sizes:0.000612999, 0 . . . Proof of Corollary 1.3.
Let ` di =1 I i be a partition of the interval [0 , M α ] into small-est possible number of connected parts such that f | I i = ˆ f i | I i for some linear func-tions ˆ f i : x ∈ R p i q x + k i ∈ R , 1 ≤ i ≤ d , where 0 = p i ∈ Z , q ∈ Z > and k i ∈ R .Let n i ≤ N i be uniquely defined integers such that mα ∈ I i iff n i < m ≤ N i , MANISH MISHRA AND AMY BINNY PHILIP ≤ i ≤ d . The result then follows from Theorem 1.1 by putting N = M and {−} λ = {−} ′∞ . (cid:3) Acknoledgement
The authors would like to thank Deepa Sahchari for helpful discussions and TianAn Wong for pointing out the reference [3]. They would especially like to thankthe anonymous referee for pointing out a serious error in an earlier draft of thisarticle because of which the statements of Theorem 1.1 and Corollary 1.3 had tobe modified.
References [1] Antal Balog, Andrew Granville, and Jozsef Solymosi,
Gaps between fractional parts, and ad-ditive combinatorics , Q. J. Math. (2017), no. 1, 1–11, DOI 10.1093/qmath/hav012.[2] FRK Chung and RL Graham, On the set of distances determined by the union of arithmeticprogressions , Ars Combinatoria (1976), 57–76.[3] Henk Don, On the distribution of the distances of multiples of an irrational number to thenearest integer , Acta Arith. (2009), no. 3, 253–264, DOI 10.4064/aa139-3-4.[4] Frank M Liang,
A short proof of the 3d distance theorem , Discrete mathematics (1979),no. 3, 325–326.[5] Jens Marklof and Andreas Str¨ombergsson, The three gap theorem and the space of lattices ,Amer. Math. Monthly (2017), no. 8, 741–745, DOI 10.4169/amer.math.monthly.124.8.741.[6] Vera T S´os,
On the theory of diophantine approximations I (on a problem of A. Ostrowski) ,Acta Mathematica Hungarica (1957), no. 3-4, 461–472.[7] Vera T S´os, On the distribution mod 1 of the sequence nα , Ann. Univ. ¨Ecient. Budapest E¨otv¨os¨Eect. Math, (1958), 127–134.[8] J´anos Sur´anyi, ¨Uber die Anordnung der Vielfachen einer reellen Zahl mod 1 , Ann. Univ. Sci.Budupest E¨otv¨s Sect. Math (1958), 107–111.[9] S ´Swierczkowski, On successive settings of an arc on the circumference of a circle , FundamentaMathematicae (1958), 187–189. Indian Institute of Science Education and Research Pune, Dr. Homi Bhabha Road,Pasha, Pune 411008, Maharashtra, India
E-mail address ::