arXiv:1302.0917v1 [math.NT] 5 Feb 2013 Conjecture. netgt odtoson conditions investigate lsia otx fsseso w iercnrec equations. congruence usu linear Remainde the two Chinese of density yields a systems which consider of formalism we context general sequel, classical the s quite In ideals, a and case. considers special rings [8] commutative Kleinert of language [7]. the in Theorem Remainder [11]. Montgomery and j < i Proof. let ytmo iercnrec qain opsesasolution. a possess 1.1 to Theorem equations congruence linear of system a a ouini n nyif only and if solution a has e od n phrases. and words Key 2010 Date pcfial,gvncollections given Specifically, stat a For directions. several in generalization admits theorem This h lsia hns eane hoe rvdsncsayads and necessary provides Theorem Remainder Chinese classical The a 1 a , . . . , ≤ eray6 2013. 6, February : ahmtc ujc Classification. Subject Mathematics e hoe . fHa[]adeecss1 3i hpe . fNiven of 2.3 Chapter in 23 – 19 exercises and [6] Hua of 7.1 Theorem See Abstract. netgt odtoson conditions investigate system pca aewhere case special a nitgrslto,adw uniytenme fsc admissib such of number the quantify we and solution, integer an has oeyRne Conjecture. Runner Lonely k . k eany be EST HNS EANE THEOREM REMAINDER CHINESE DENSITY A CieeRmidrTheorem) Remainder (Chinese ie collections Given k nees hntesse fcongruences of system the Then integers. A hns eane hoe,Dohnieapoiain oeyRun Lonely approximation, Diophantine Theorem, Remainder Chinese A and and A a B i and A ≡ oss fitraso eiu lse a plcto othe to application has classes residue of intervals of consist A B and a and B htesr ht o tlatsm ( some least at for that, ensure that j 1. .JSNGIBSON JASON D. htesr ht o tlatsm ( some least at for that, ensure that o gcd( mod 17 Piay,1J1(Secondary). 11J71 (Primary), 11B75 x x x B B Introduction ≡ ≡ ≡ x x frsdecassmodulo classes residue of frsdecassmodulo classes residue of . . . ≡ ≡ a a a 1 2 k a b mod mod mod 1 mod mod . m Let i n m , m m m m 1 2 k j m ) 1 o l ar findices of pairs all for m , . . . , m k be and m hoe rmdi the in framed Theorem r k epis( pairs le ,b a, n and ffiin odtosfor conditions ufficient oiieitgr,and integers, positive epciey we respectively, , ) e .. Hungerford e.g., ee, mn faChinese a of ement ,b a, B × A ∈ n lsaeeta a as statement al epciey we respectively, , ,b a, ) B × A ∈ .The ). Zuckerman, , ,j i, the , with the , 1 ner ≤ system x ≡ a mod m (1.1) x ≡ b mod n has an integer solution, and we quantify the number of such admissible pairs (a, b). For instance, if the collections A and B satisfy |A| = |B| = 1, then the Chinese Remainder Theorem (Theorem 1.1) provides appropriate conditions, namely, that the greatest common divisor gcd(m, n) divides the difference a − b. For fixed m and n, if the collections A and B are not large enough (in a suitable sense), then it might occur that, because the gcd condition doesn’t hold, the system (1.1) admits no solution with (a, b) ∈ A×B. Still, one expects that requiring A and B to have large enough density in comparison to the full set of residue classes modulo m and n, respectively, will force the gcd condition to hold, so that the classical Chinese Remainder Theorem then yields a solution (or solutions) to the system of equations. We make this intuition precise in Theorems 2.1 and 2.2, establishing a density Chinese Remainder Theorem for the case of arbitrary collections of residue classes and for collections of intervals of residue classes, respectively. Results of this form, particularly generalizations of Corollary 2.4, have application to the Lonely Runner Conjecture, which we also briefly discuss.
2. Intersections of sets of arithmetic progressions In this section, we state our main results and some corollaries. The proofs of Theorems 2.1 and 2.2 appear in the following two sections. Theorem 2.1 (Density CRT). Let A and B be collections of residue classes modulo m and n, respectively. Let g = gcd(m, n), and let h denote the number of solutions to the linear system x ≡ a mod m (2.1) x ≡ b mod n modulo mn/g as (a, b) ranges over A × B. Write m m |A| = A + r , 0 ≤ r < , g A A g n n (2.2) |B| = B + r , 0 ≤ r < . g B B g Then h, the number of solutions to the linear system (2.1) modulo mn/g, satisfies
0, if A + B
h ≥ ABg + ArB + BrA + max(0,rA + rB − g). (2.6) Corollary 2.3 (gcd(m, n) = 1). Given the conditions of Theorem 2.2 with the additional assumption that gcd(m, n)=1, then the linear system (2.4) possesses exactly |A||B| solu- tions. Proof. This follows immediately from Theorem 2.2. Also, more directly, note that the condition gcd(m, n) = 1 ensures that each choice (a, b) ∈ A × B of residue classes leads to a solution modulo mn of the linear system by way of the classical Chinese Remainder Theorem (Proposition 1.1). Corollary 2.4 (Density statement). Given the conditions of Theorem 2.2, with the addi- tional assumptions that 1 |A| > m, 3 (2.7) 1 |B| > n, 3 and that m and n are distinct, then the linear system (2.4) possesses a solution. 1 Also, the constant 3 can not be taken to be smaller while keeping the above conclusion of this corollary. Proof. Following Theorem 2.2, write
|A| = Ag + rA, 0 ≤ rA < g, (2.8) |B| = Bg + rB, 0 ≤ rB < g, with g = gcd(m, n). To establish the desired claim of this corollary, we must examine the terms appearing on the right side of (2.6). If AB > 0, then the corollary follows immediately, and so it remains to consider the possibility that AB = 0. To that end, suppose that A = 0 and B = 0. (If B > 0, then the BrA term in (2.6) is 1 m nonzero, and again we are done.) Then g>rA = |A| > 3 m implies that g < 3. This forces m g to be either 1 or 2. In the first case, we have m = g. Since m and n are distinct and g must divide n, it must be that n = mn0, where the integer n0 satisfies n0 ≥ 2. Then we have 1 1 2 2 |B| > n = mn ≥ m = g. (2.9) 3 3 0 3 3 1 1 2 With rA > 3 m = 3 g and rB = |B| > 3 g, this establishes that the rA + rB − g term of (2.6) is nonzero, and then the linear system posseses a solution by Theorem 2.2. 3 m In the second case, that g = 2, we have m =2g. Since m and n are distinct and g must divide n, it must be that n = gn0, where the integer n0 satisfies n0 =1 or n0 ≥ 3. If n0 = 1, then m =2g and n = g, and a solution exists by the reasoning above. If n0 ≥ 3, then 1 1 |B| > n = gn ≥ g (2.10) 3 3 0
implies that B > 0, and, consequently, the BrA term in (2.6) is nonzero. Thus, in the case also, a solution again exists by Theorem 2.2. 1 Finally, to see that the constant 3 is optimal, let M be a large integer, and set m =3M, n =2 · 3M. Let A = {0, 1,...,M − 1} (2.11) and B = {M, M +1,..., 3M − 1}. (2.12) 1 1 Then gcd(m, n) = gcd(3M, 2 · 3M)=3M, with |A| = M = 3 m and |B| = 2M = 3 n. If (a, b) ∈A×B, then a − b is nonzero modulo gcd(m, n)=3M, so that the classical Chinese Remainder Theorem (Proposition 1.1) implies that the linear system has no solution.
3. Arbitrary collections In this section, we prove Theorem 2.1. Given collections A and B of residue classes modulo m and n, respectively, the classical Chinese Remainder Theorem reduces counting the number of solutions of the linear system (2.1) to counting the pairs (a, b) ∈A×B with a ≡ b mod gcd(m, n). To that end, let g = gcd(m, n), and, for a collection of residue classes C and integer i with 1 ≤ i ≤ g, partition the collection C into sets
Ci = {c ∈ C : c ≡ i mod g}, (3.1) and define the counting function f by
f(i, C)= |Ci|. (3.2) The Chinese Remainder Theorem then yields that the number of solutions to the linear system (2.1) is given by the sum
g f(i, A)f(i, B). (3.3) X i=1 Note that we have g |A| = f(i, A), X i=1 g (3.4) |B| = f(i, B). X i=1 4 Moreover, for 1 ≤ i ≤ g, the right-hand summands in (3.4) are each bounded above, with m f(i, A) ≤ , g n (3.5) f(i, B) ≤ . g To establish Theorem 2.1, we must find a lower bound on (3.3) under the conditions (3.4) and (3.5). We do so in three steps. First, we use an inequality on rearrangements, Lemma 3.1, to reduce to the case where the values f(i, A) are increasing and the values f(i, B) are decreasing. Next, in this case, Lemma 3.2 yields an extremal distribution for these values under the constraints of (3.4) and (3.5). Finally, Lemma 3.3 provides a case analysis for the explicit evaluation of the sum (3.3) in this extremal case. Theorem 2.1 follows directly from these three lemmas.
Lemma 3.1 (Rearrangement inequality). For each pair of ordered real sequences a1 ≤ a2 ≤ ···≤ an and b1 ≤ b2 ≤···≤ bn, and for each permutation σ :[n] → [n], we have n n n a b ≤ a b ≤ a b . (3.6) X k n−k+1 X k σ(k) X k k k=1 k=1 k=1 Proof. See Chapter 5 of Steele [12] and Chapter X of Hardy, Littlewood, and P´olya [5]. Lemma 3.2 (Extremal distribution). For each pair of non-negative ordered real sequences n n a1 ≤ a2 ≤···≤ an and b1 ≤ b2 ≤···≤ bn satisfying A = ak, B = bk, and, for Pk=1 Pk=1 all k with 1 ≤ k ≤ n, the bounds ak ≤ qA and bk ≤ qB, then we have n n a∗b∗ ≤ a b , (3.7) X k n−k+1 X k n−k+1 k=1 k=1 where 0, if k < n − ⌊A/q ⌋, A a∗ = A − q ⌊ A ⌋, if k = n − ⌊A/q ⌋, (3.8) k A qA A qA, if k > n − ⌊A/qA⌋, and 0, if k < n − ⌊B/q ⌋, B b∗ = B − q ⌊ B ⌋, if k = n − ⌊B/q ⌋, (3.9) k B qB B qB, if k > n − ⌊B/qB⌋. Proof. Since b1 ≤ b2 ≤···≤ bn, we have n n a∗b ≤ a b . (3.10) X k n−k+1 X k n−k+1 k=1 k=1 ∗ ∗ ∗ Then, because a1 ≤ a2 ≤···≤ an, we have n n a∗b∗ ≤ a∗b , (3.11) X k n−k+1 X k n−k+1 k=1 k=1 and the result follows. 5 Lemma 3.3 (Extremal sum). Using the notation of Lemma 3.2, if
s = ⌊A/qA⌋ + ⌊B/qB⌋ +1,
rA = A − qA⌊A/qA⌋, (3.12)
rB = B − qB⌊B/qB⌋, then we have
n 0, if s < n, ∗ ∗ a b = rArB, if s = n, (3.13) X k n−k+1 k=1 (s − n − 1)qAqB + rAqB + rBqA, if s > n. Proof. First, note that (3.9) yields that q , if k < ⌊B/q ⌋ +1, B B b∗ = B − q ⌊ B ⌋, if k = ⌊B/q ⌋ +1, (3.14) n−k+1 B qB B 0, if k > ⌊B/qB⌋ +1. ∗ ∗ Then, (3.8) and (3.14) together imply that the summands akbn−k+1 in (3.13) vanish unless k satisfies n − ⌊A/qA⌋ ≤ k ≤ ⌊B/qB⌋ +1. (3.15) This allows the sum in (3.13) to be written as
n ⌊B/qB ⌋+1 a∗b∗ = a∗b∗ . (3.16) X k n−k+1 X k n−k+1 k=1 k=n−⌊A/qA⌋ The analysis of this sum now requires examining three cases: s < n, s > n, and s = n. If
s = ⌊A/qA⌋ + ⌊B/qB⌋ +1 < n, (3.17) then the sum is empty. If s > n, then the sum consists of a first term, last term, and s−n−1 middle terms. The first and last terms contribute rAqB and rBqA, respectively, while the remaining terms each have identical value qAqB. Finally, if s = n, the sum consists of a single term of value rArB. 4. Collections of intervals In this section, we prove Theorem 2.2. As in the previous section, given collections A and B of residue classes modulo m and n, respectively, the classical Chinese Remainder Theorem reduces counting the number of solutions of the linear system (2.4) to counting the pairs (a, b) ∈ A×B with a ≡ b mod gcd(m, n). Unlike in the previous section, the fact that the collections of residue classes in Theorem 2.2 consist of intervals of classes, rather than arbitrary sets of classes, substantially simplifies the analysis of this counting problem. Proof of Theorem 2.2. With g = gcd(m, n), we have
|A| = Ag + rA, 0 ≤ rA < g, (4.1) |B| = Bg + rB, 0 ≤ rB < g. It remains to count the pairs (a, b) ∈A×B with a ≡ b mod g. Since A consists of intervals of residue classes, it follows that A can be partitioned into A sub-intervals of length g, each sub-interval containing g distinct elements modulo g, along 6 with a smaller sub-interval of length rA, containing rA < g distinct elements modulo g. Similarly, B can be partitioned into B sub-intervals of length g, each sub-interval containing g distinct elements modulo g, along with a smaller sub-interval of length rB, containing rB
ABg + ArB + BrA + min(0,rA + rB − g) (4.2) solutions, completing the proof of Theorem 2.2.
5. The Lonely Runner Conjecture The Lonely Runner Conjecture, having its origins in view-obstruction problems and in diophantine approximation, seems to be due independently to Wills [13] and Cusick [3]. A problem in n-dimensional geometry view obstruction motivated Cusick’s statement of the problem, while Wills viewed the question from the perspective of Diophantine approximation. Let k ≥ 2 be an integer, and let m1,...,mk be distinct positive integers. For x ∈ R, let kxk denote the distance from x to the integer nearest, i.e., kxk = min{|x − n| : n ∈ Z}. (5.1) Montgomery includes the Wills version as Problem 44 in the Diophantine Approximation section of the appendix of unsolved problems in [10].
Conjecture 5.1. (Wills) If 1 ≤ m1 < m2 <...
with analogous statements holding for TQ(n) and B. Thus, Corollary 2.4 applies, yielding a solution to the linear system of congruences and hence the existence of a time belonging to both TQ(m) and TQ(n).
6. Further work First, as with the classical Chinese Remainder Theorem, results for systems containing more than two linear congruence equations would be both interesting and useful. In partic- ular, such a generalization in the case of collections of intervals can be applied to the Lonely 1 Runner Conjecture. For instance, the constant 3 appearing in Corollary 2.4 is the analogue of the same constant appearing in the Lonely Runner Conjecture with k = 2 runners. One might hope for such similarities to continue for larger values of k. Next, other possibilities for the collections A and B, for example, random collections (for suitable notions of random), collections nicely distributed among residue classes, and collections with other arithmetic structure might be amenable to analysis. 8 Finally, regardless of the number of linear congruence equations or the types of collec- tions involved, structural information beyond mere existence and quantity for the admissible classes and the resulting solutions could be useful in iterated applications of this, and other, density Chinese Remainder Theorems. References [1] U. Betke and J. M. Wills. Untere Schranken f¨ur zwei diophantische Approximations-Funktionen. Monatsh. Math., 76:214–217, 1972. [2] Wojciech Bienia, Luis Goddyn, Pavol Gvozdjak, Andr´as Seb˝o, and Michael Tarsi. Flows, view obstruc- tions, and the lonely runner. J. Combin. Theory Ser. B, 72(1):1–9, 1998. [3] T. W. Cusick. View-obstruction problems in n-dimensional geometry. J. Combinatorial Theory Ser. A, 16:1–11, 1974. [4] T. W. Cusick and Carl Pomerance. View-obstruction problems. III. J. Number Theory, 19(2):131–139, 1984. [5] G. H. Hardy, J. E. Littlewood, and G. P´olya. Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988. Reprint of the 1952 edition. [6] Loo Keng Hua. Introduction to number theory. Springer-Verlag, Berlin, 1982. Translated from the Chi- nese by Peter Shiu. [7] Thomas W. Hungerford. Algebra, volume 73 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1980. Reprint of the 1974 original. [8] Ernst Kleinert. Some remarks on the Chinese remainder theorem. Arch. Math. (Basel), 52(5):433–439, 1989. [9] Daphne Der-Fen Liu. From rainbow to the lonely runner: a survey on coloring parameters of distance graphs. Taiwanese J. Math., 12(4):851–871, 2008. [10] Hugh L. Montgomery. Ten lectures on the interface between analytic number theory and harmonic analysis, volume 84 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1994. [11] Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery. An introduction to the theory of numbers. John Wiley & Sons Inc., New York, fifth edition, 1991. [12] J. Michael Steele. The Cauchy-Schwarz master class. MAA Problem Books Series. Mathematical Asso- ciation of America, Washington, DC, 2004. An introduction to the art of mathematical inequalities. [13] J¨org M. Wills. Zwei S¨atze ¨uber inhomogene diophantische Approximation von Irrationalzahlen. Monatsh. Math., 71:263–269, 1967. [14] Xuding Zhu. Circular chromatic number of distance graphs with distance sets of cardinality 3. J. Graph Theory, 41(3):195–207, 2002.
Department of Mathematics and Statistics, Eastern Kentucky University, KY 40475, USA E-mail address: [email protected]
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