Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing , People s Republic of China. Received 8 July 2005; accepted 2 February 2006

Transcription

1 Adv in Appl Math , no 2, A CONNECTION BETWEEN COVERS OF THE INTEGERS AND UNIT FRACTIONS Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing 20093, People s Republic of China Received 8 July 2005; accepted 2 February 2006 Abstract For integers a and n > 0, let an denote the residue class {x Z : x a mod n} Let A be a collection {a s } k s of finitely many residue classes such that A covers all the integers at least m times but {a s } k s does not We show that if n k is a period of the covering function w A x { s k : x a s } then for any r 0,, n k there are at least m integers in the form / r/n k with I {,, k } MSC: primary B25; secondary B75, D68 Introduction For an integer a and a positive integer n, we use an to denote the residue class {x Z : x a mod n} For a finite system A {a s } k s of residue classes, the function w A : Z {0,, } given by w A x { s k : x a s } is called the covering function of A Clearly w A x is periodic modulo the least common multiple N A of the moduli n,, n k, and it is easy to verify the following well-known equality: N A N A x0 w A x k s address: zwsun@njueducn URL: Supported by the National Science Fund for Distinguished Young Scholars no and a Key Program of NSF no in China

2 2 ZHI-WEI SUN As in [7] we call ma min x Z w A x the covering multiplicity of A For example, B {02, 03, 4, 56, 72} has covering multiplicity mb, because the covering function is periodic modulo N B 2, and { if x {, 2, 3, 4, 7, 8, 0, }, w B x 2 if x {0, 5, 6, 9} Let m be a positive integer If w A x m for all x Z, then we call A an m-cover of the integers, and in this case we have the well-known inequality k s / m The term -cover is usually replaced by the word cover If A is an m-cover of the integers but A t {a s } s [,k]\{t} is not where [a, b] {x Z : a x b} for a, b Z, then we say that A forms an m-cover of the integers with a t n t irredundant For example, {02, 2, 23} is a cover of the integers in which 23 is redundant while 02 and 2 are irredundant If w A x m for all x Z, then A is said to be an exact m-cover of the integers, and in this case we have the equality k s / m By a graph-theoretic argument, M Z Zhang [2] showed that if m > then there are infinitely many exact m-covers of the integers each of which cannot be split into an n-cover and an m n-cover of the integers with 0 < n < m Covers of the integers by residue classes were first introduced by P Erdős cf [] in the 930s, who observed that the system B mentioned above is a cover of the integers with distinct moduli The topic of covers of the integers has been an active one in combinatorial number theory cf [3, 4], and many surprising applications have been found see, eg, [, 2, 9, ] The so-called m-covers and exact m-covers of the integers were systematically studied by the author in the 990s Concerning the cover B given above one can easily check that { n S n { 0, 2,, 2 : S {2, 3, 4, 6, 2} } } { + r 2 : r 0,, 2, 3, 4 } This suggests that for a general m-cover of the integers we should investigate the set { / : I [, k]} In this paper we establish the following new connection between covers of the integers and unit fractions Theorem Let A {a s } k s be an m-cover of the integers with the residue class a k n k irredundant If the covering function w A x is

3 COVERS OF THE INTEGERS AND UNIT FRACTIONS 3 periodic modulo n k, then for any r 0,, n k we have { { : I [, k ] and } r } m, 2 n k where α and {α} denote the integral part and the fractional part of a real number α respectively Note that n k in Theorem needn t be the largest modulus among n,, n k In the case m and n k N A, Theorem is an easy consequence of [5, Theorem ] as observed by the author s twin brother Z H Sun When w A x m for all x Z, the author [6] even proved the following stronger result: { I [, k ] : a } n k m a/n k for all a 0,, Given an m-cover {a s } k s of the integers with a k n k irredundant, by refining a result in [7] the author cahow that there exists a real number 0 α < such that 2 with r/n k replaced by α + r/n k holds for every r 0,, n k Here we mention two local-global results related to Theorem a Z W Sun [5] {a s } k s forms an m-cover of the integers if it covers {{ /} : I [, k]} consecutive integers at least m times b Z W Sun [0] {a s } k s is an exact m-cover of the integers if it covers k s {r/ : r [0, ]} consecutive integers exactly m times Corollary Suppose that the covering function of A {a s } k s has a positive integer period n 0 If there is a unique a 0 [0, n 0 ] such that w A a 0 ma, then for any D Z with D ma we have {{ } : I [, k] and } { } r D : r [0, n 0 ] n 0 Proof Let m ma + Clearly A {a s } k s0 forms an m-cover of the integers with a 0 n 0 irredundant As w A x w A x is the characteristic function of a 0 n 0, w A x is also periodic mod n 0 Applying Theorem we immediately get the desired result We will prove Theorem in Section 3 with help from some lemmas given in the next section

4 4 ZHI-WEI SUN 2 Several Lemmas Lemma Let be a finite system of residue classes with ma m, and let m,, m k be any integers If fx,, x k is a polynomial with coefficients in the complex field C and deg f m, then for any z Z we have I f[ I ],, [k I ]e 2πi a s zm s / I [,k] k ci z s [,k]\i z e 2πia s zm s /, where [s I ] takes or 0 according as s I or not, I z { s k : z a s }, and ci z [ z x s ]fx,, x k is the coefficient of the monomial z x s in fx,, x k Proof Write fx,, x k j,,j k 0 c j,,j k x j x j k k Observe that I [,k] j,,j k 0 j + +j k m j,,j k 0 j + +j k m I f[ I ],, [k I ]e 2πi a s zm s / c j,,j k c j,,j k I [,k] 3 k [s I ] j s I e 2πi a s zm s / s Jj,,j k I [,k] I e 2πi a s zm s /, where Jj,, j k { s k : j s 0} Let j,, j k be nonnegative integers with j + + j k m If I z Jj,, j k, then Jj,,j k I [,k] I e 2πi a s zm s / 0 since I e 2πi a s zm s / I [,k]\jj,,j k s [,k]\jj,,j k e 2πia s zm s / 0 If I z Jj,, j k, then m ma I z Jj,, j k j + + j k m;

5 COVERS OF THE INTEGERS AND UNIT FRACTIONS 5 hence I z Jj,, j k and j s for all s I z Combining the above we find that the left-hand side of 3 coincides with ci z I z I [,k] I e 2πi a s zm s / ci z I z e 2πi z a s zm s / e 2πia s zm s / k ci z s [,k]\i z s [,k]\i z e 2πia s zm s / This proves the desired equality 3 Lemma 2 Let be an m-cover of the integers with a k n k irredundant, and let m,, m k be positive integers Then, for any 0 α < we have C 0 α C nk α, where C r α with r [0, n k ] denotes the sum I [,k ] { m s/ }α+r/n k I m s/ e 2πi a s a k m s / m Proof This follows from [7, Lemma 2] Lemma 3 Let be an m-cover of the integers with a k n k irredundant Suppose that n k is a period of the covering function w A x Then, for any z a k n k we have s [,k]\i z e 2πia s z/ n k e 2πit a wa t m k/n k z t where I z { s k : z a s } Proof Since a k n k is irredundant, we have w A z 0 m for some z 0 a k n k As the covering function of A is periodic mod n k, I z w A z m for all z a k n k Now fix z a k n k Since w A x is periodic modulo n k, by [8, Lemma 2] we have the identity k n k y N/ e 2πia s/ s t y N/n k e 2πit/n k wa t,

6 6 ZHI-WEI SUN where N N A is the least common multiple of n,, n k Putting y r /N e 2πiz/N with r 0, we then get k n k r / e 2πia s z/ s t r /n k e 2πit z/n k wa t Therefore e 2πia s z/ s [,k]\i z lim r s [,k]\i z lim r r / e 2πia s z/ nk t r/n k e 2πit a k/n k w At z r / e 2πia s z/ nk t r/n k e 2πit a k/n k w At r m r lim lim r r / r z n k lim r /n k e 2πit a k/n k wat m, r z t and hence the desired result follows 3 Proof of Theorem In the case k, we must have m and n k ; hence the required result is trivial Below we assume that k > Let r 0 [0, n k ] and D {d n + r 0 /n k : n [, m ]}, where d,, d m are m distinct nonnegative integers If m then we set D We want to show that there exists an I [, k ] such that { /} r 0 /n k and / D Define fx,, x k x + + x k d n n k d D An empty product is regarded as Then deg f D m For any z a k n k, the set I z { s k : z a s } has cardinality m since a k n k is irredundant and w A x is periodic mod n k Observe that the coefficient [ c z z \{k} [ x s ]fx,, x k z \{k} ] k m x s x s s

7 COVERS OF THE INTEGERS AND UNIT FRACTIONS 7 coincides with m!/ z \{k} by the multinomial theorem For I [, k ] we set vi f[ I ],, [k I ] As { s k : x a s } deg f for all x Z, in view of Lemmas and 3 we have I [,k ] k c z I vie 2πi a s z/ s [,k ]\I z m m! z \{k} e 2πia s z/ n k z t e 2πit a k/n k wa t m C, where C is a nonzero constant not depending on z a k n k By the above, N A N A C and hence where C r x0 C I [,k ] I [,k ] I vie 2πi a s a k n k x/ I vie 2πi N A a s a k / I [,k ] n k / Z I [,k ] { /}r/n k Let r [0, n k ] Write P r x d D x0 I vie 2πi a s a k / I d D e 2πix n k/ n k r0 C r, d e 2πi a s a k / x + r m x d c n,r n k n n0

8 8 ZHI-WEI SUN where c n,r C By comparing the leading coefficients, we find that c m,r m! Observe that C r I [,k ] { /}r/n k m c n,r n0 c m,r I P r I [,k ] { /}r/n k I [,k ] { /}r/n k e 2πi a s a k / I /n s n e 2πi a s a k / I /n s e 2πi a s a k / ; m in taking the last step we note that if 0 n < m then I /n s e 2πi a s/ 0 n I [,k ] { /}r/n k by [5, Theorem ] since {a s } k s By Lemma 2 and the above, C r m! I [,k ] { /}0 does not depend on r [0, n k ] Combining the above we obtain that n k C r0 is an m -cover of the integers I /n s m n k r0 C r C 0 e 2πi a s a k / So there is an I [, k ] for which { /} r 0 /n k, / D and hence / {d n : n [, m ]} This concludes our proof References P Erdős, On integers of the form 2 k + p and some related problems, Summa Brasil Math M Filaseta, Coverings of the integers associated with an irreducibility theorem of A Schinzel, in: M A Bennett, B C Berndt, N Boston, H G Diamond, A J Hildebrand, W Philipp Eds, Number Theory for the Millennium Urbana, IL, 2000, vol II, pp -24, A K Peters, Natick, MA, 2002

9 COVERS OF THE INTEGERS AND UNIT FRACTIONS 9 3 R K Guy, Unsolved Problems in Number Theory, third ed, Springer, New York, 2004 Sections F3 and F4 4 Š Porubský and J Schönheim, Covering systems of Paul Erdös: Past, present and future, in: G Halász, L Lovász, M Simonvits, V T Sós Eds, Paul Erdös and his Mathematics, I, Bolyai Soc Math Stud, vol, 2002, pp Z W Sun, Covering the integers by arithmetic sequences, Acta Arith Z W Sun, Exact m-covers and the linear form k s x s/, Acta Arith Z W Sun, On covering multiplicity, Proc Amer Math Soc Z W Sun, On the function wx { s k : x a s mod }, Combinatorica Z W Sun, Unification of zero-sum problems, subset sums and covers of Z, Electron Res Announc Amer Math Soc Z W Sun, Arithmetic properties of periodic maps, Math Res Lett Z W Sun, A local-global theorem on periodic maps, J Algebra M Z Zhang, Irreducible systems of residue classes that cover every integer exactly m times, Sichuan Daxue Xuebao Nat Sci Ed