Journal of Number Theory

Transcription

1 Journal of Number Theory Contents lsts avalable at ScVerse ScenceDrect Journal of Number Theory Algebrac numbers, hyperbolcty, and densty modulo one A. Gorodnk,S.Kadyrov School of Mathematcs, Unversty of Brstol, Brstol BS8 1TW, UK artcle nfo abstract Artcle hstory: Receved 1 September 2011 Revsed 2 May 2012 Accepted 2 May 2012 Avalableonlne15July2012 Communcated by Davd Goss Keywords: Densty modulo one Multplcatvely ndependent numbers Compact abelan group Hgher-rank abelan acton We prove the densty of the sets of the form { λ m 1 μn 1 ξ 1 + +λ m k μn k ξ k: m, n N modulo one, where λ and μ are multplcatvely ndependent algebrac numbers satsfyng some addtonal assumptons. The proof s based on analysng dynamcs of hgher-rank actons on compact abelan groups Elsever Inc. All rghts reserved. 1. Introducton The am of ths paper s to generalse the followng theorem of B. Kra [4]: Theorem 1.1. Let p, q 2, = 1,...,k, be ntegers such that a each par p, q s multplcatvely ndependent, 1 b for all j, p, q p j, q j. Then for all real numbers ξ,= 1,...,k, wth at least one of ξ s rratonal, the set { k p n qm ξ : m,n N =1 s dense modulo one. * Correspondng author. E-mal addresses: a.gorodnk@brstol.ac.uk A. Gorodnk, shral.kadyrov@brstol.ac.uk S. Kadyrov. 1 Aparλ, μ s called multplcatvely ndependent f λ m μ n for all m, n Z 2 \{0, X/$ see front matter 2012 Elsever Inc. All rghts reserved.

2 2500 A. Gorodnk, S. Kadyrov / Journal of Number Theory We prove an analogous results wth p and q beng algebrac numbers. For ths we need to ntroduce the noton of hyperbolcty. For any prme p ncludng p =, let Q p denote the algebrac closure of Q p.inpartcular,q = C. A semgroup Σ consstng of algebrac numbers wll be called hyperbolc provded that for every prme p ncludng p =, f there s a feld embeddng θ : QΣ Q p such that θσ { z p 1, then for all feld embeddngs θ : QΣ Q p,wehave θσ { z p = 1. For example, f α > 1 s a real algebrac nteger, then the semgroup α s hyperbolc provded that none of the Galos conjugates of α have absolute value one. The algebrac closure of the feld of ratonal numbers Q wll be denoted by Q. Our man result s the followng: Theorem 1.2. Let λ, μ,= 1,...,k, be real algebrac numbers satsfyng λ, μ > 1 such that a each par λ, μ s multplcatvely ndependent, b for all j, θ Gal Q/Q, andu N, θλ u,θμ u λ u j, μu j, c each semgroup λ, μ s hyperbolc. Then for all real numbers ξ,= 1,...,k, wth at least one of ξ satsfyng ξ / Qλ, μ,theset { k λ n μm ξ : m,n N =1 s dense modulo one. Prevously, D. Berend [3] have nvestgated the case k = 1, and R. Urban [5 7] have proved several partal results when k = 2. In the next secton, we ntroduce a compact abelan group Ω equpped wth an acton of a commutatve semgroup Σ and show that the sequence that appears n the man theorem s closely related to a sutably chosen orbt Σω n Ω. More precsely, ths sequence s obtaned by applyng a projecton map Π : Ω R/Z. Ths constructon s analogous to the one of Berend n [3], but n the case k > 1, we have to deal wth a larger space Ω where the structure of orbts of Σ s not well understood, and ths requres several addtonal arguments. The dea of the proof s to show that the closure Σω has an addtonal structure. In Secton 3 we show that Σω contans a torson pont. We note that the hyperbolcty assumpton c s necessary for exstence of a torson pont. Then usng a lmtng argument n a neghbourhood of ths torson pont, we demonstrate n Secton 4 that Σω approxmates arbtrary long lne segments. Fnally, we complete the proof n Secton 5 by showng that the projectons under Π of such lne segments cover R/Z. Ths s where the ndependence assumpton b s used.

3 A. Gorodnk, S. Kadyrov / Journal of Number Theory Settng In ths secton, we construct a compact abelan group Ω and a commutatve semgroup Σ of epmorphsms of Ω. We show that there s a natural projecton map Π : Ω R/Z, and for a sutably chosen ω Ω, { k ΠΣω = λ m μ n ξ : m,n N =1 mod 1. 1 Ths reduces the proof of the theorem to analyss of orbt structure of Σ n Ω. Now we explan the detals of ths constructon. Let K be a number feld. We fx a bass β 1,...,β r of the rng of algebrac ntegers of K. To every element α K we assocate a matrx Mα = a jl Mat r Q determned by r α β j = a jl β l, 1 j r. 2 l=1 Suppose that Mα Mat r Z[1/a] for some a N, and s mnmal wth ths property. We set Ω r a := Rr p a Q r p, Ω r a = Ω r a /Z[1/a]r, where Z[1/a] r s embedded n Ω a r by the map z z, z,..., z. ThenΩr s a compact abelan group. Every matrx M Mat r Z[1/a] naturally acts on dagonally and defnes a map Ω r a M : Ω r a Ωr a. The dstrbuton of orbts of such maps wll play a crucal role n ths paper. The followng lemma wll be useful: Lemma 2.1. If a prme p dvdes a, then there s an embeddng θ : Qα Q p such that θα p > 1. Proof. We wrte a = p n b wth gcdp, b = 1 and set β = bα. It follows from 2 that for every Galos conjugate θβ, the multplcaton by p n θβ preserves the ntegral module Zθβ 1 + +Zθβ r. Therefore, p n θβ s an algebrac nteger, and θβ q 1 for all Galos conjugates of β and all prmes q p. Suppose that also θβ p 1 for all Galos conjugates of β. Thenβ s an algebrac nteger and, n partcular, β β j Zβ 1 + +Zβ r for all j. On the other hand, snce s mnmal wth the property Mα Mat r Z[1/a], t follows that β β j / Zβ 1 + +Zβ r for some j. Ths contradcton shows that θα p = θβ p > 1 for some θ, asrequred.

4 2502 A. Gorodnk, S. Kadyrov / Journal of Number Theory Now we adopt ths constructon to our settng. Let K be a number feld of degree r that contans λ and μ, and let A = Mλ and B = Mμ be the matrces n Mat r Z[1/ ] defned as above, where N s mnmal wth ths property. We denote by Σ the commutatve semgroup generated by A and B. Ths semgroup acts on Ω r and Ω r. We also consder the semgroup Σ := { A n 1 Bm 1,...,An k Bm k : m,n N generated by A := A 1,...,A k and B := B 1,...,B k that naturally acts on Ω := k =1 Ω r. We denote by π : Ω := k =1 Ω r Ω the correspondng projecton map. We wrte Ω = h k =1 j=1 where p 1 =,...,p h are the prmes dvdng herewewrteq = R. We denote by {e jl the standard bass of Ω, and ntroduce a projecton map Q r p j Π : Ω R/Z : s jl e jl {s j1 pj mod 1, 3, j,l, j where {x denotes the usual fractonal part, and {x p denotes the p-adc fractonal part. Namely, for x = u= N x u p u Q p,weset{x p = 1 u= N x u p u. It s easy to check Π s contnuous, and k Π Z[1/ ] r = 0 mod 1. =1 Hence, Π also defnes a map Ω R/Z. It follows from the defnton of A = Mλ and B = Mμ that they have a jont egenvector v R r wth egenvalues λ and μ respectvely. Let us assume for now that the frst coordnate of v s nonzero. Then we normalse v so that ths coordnate s one. We set v = k ξ v, 0,...,0 Ω and ω = πv Ω. =1 Then t follows from the defnton of Π that 1 holds. Although ths constructon may be appled to any choces of the number felds K, t s most convenent to choose these felds to be of the smallest sze, and we adopt an dea from [1]. For every = 1,...,k, wepckl N so that Qλ l, μl = l=1 Qλl, μl, and we set l 0 = k =1 l. Then Qλ l 0, μ l 0 = l=1 Qλl, μl. Weobservethatthenumbersλl 0 and μ l 0 are satsfyng the assumptons Theorem 1.2, and f we prove the clam of the theorem for these numbers, then the theorem would follow for λ s and μ s as well. Hence, from now on we assume that l 0 = 1 and take K = Qλ, μ.

5 A. Gorodnk, S. Kadyrov / Journal of Number Theory The man advantage of ths constructon s the followng lemma: Lemma 2.2. There exsts C Σ such that the characterstc polynomal of C u u N. s rreducble over Q for every Proof. Ths follows from [1, Lemma 4.2]. Indeed, snce Qλ, μ = l=1 Qλl, μl, by ths lemma there exsts σ n the semgroup generated by λ and μ such that Qσ n = Qλ, μ for all n N. Snce the matrx C n = Mσ n Mat r Z[1/ ] has an egenvalue σ n of degree r over Q, the clam follows. We denote by v l,1 l r, the egenvectors of the matrx C. Snce all the egenvalues of C are dstnct, t follows that v l s are also egenvectors of the whole semgroup Σ.ForD Σ,wedenote by λ l D the correspondng egenvalue. In partcular, we set λ l = λ l A and μ l = λ l B. We choose the ndces, so that λ 1 = λ and μ 1 = μ. Snce the characterstc polynomal of C s rreducble, all the egenvectors of v l,1 l r, are conjugate under the Galos acton, and t follows that ther coordnates wth respect to the standard bass are nonzero. It follows from Lemma 2.2 that λ l1 C u λ l2 C u for all l 1 l 2 and u N. Hence, n partcular, λ u l 1,μ u l 1 λ u l 2,μ u l 2 for all l 1 l 2 and u N. 4 We also ntroduce an egenbass for the space Ω. LetL j be the splttng feld of the matrx C over Q pj.weset V = h r =1 j=1 V j where V j = L r j. We denote by v jl, l = 1,...,r, the bass of the factor V j consstng of egenvectors of C chosen as above. Then v jl wth = 1,...,k, j = 1,...,h, l = 1,...,h forms a bass of V consstng of egenvectors of Σ. In these notaton, v = k ξ v 11 and ω = πv. =1 We normalse the egenvectors v jl so that ther frst coordnates wth respect the standard bases of L r j are equal to one. Then the projecton map Π s gven by 3. Exstence of torson elements Π c jl v jl = jl pj mod Z. 5, j,l{c, j,l In ths secton we nvestgate exstence of torson elements n closed Σ-nvarant subsets of Ω and prove Proposton 3.1. Every closed Σ-nvarant subset of Ω contans a torson element. We start the proof wth a lemma that generalses [1, Proposton 4.1], whch dealt wth toral automorphsms. Gven a topologcal semgroup Σ 0 actng on a topologcal space Ω 0, we say that a subset M of Ω 0 s Σ 0 -mnmal f t s non-empty, closed, Σ 0 -nvarant, and mnmal wth respect to these propertes.

6 2504 A. Gorodnk, S. Kadyrov / Journal of Number Theory Lemma 3.2. Every Σ -mnmal subset of Ω r conssts of torson elements. Proof. We consder the decomposton where V j = V 1 V >1 j j V 1 j := v jl : λ l D pj 1forallD Σ, V >1 j := v jl : λl D pj > 1 for some D Σ. In vew of Lemma 2.1, the assumpton that the semgroup Σ s hyperbolc mples that for every, j,l there exsts D Σ such that λl D pj 1. 6 Let M be a Σ -mnmal subset of Ω r. Suppose, frst, that M s fnte. We recall that the acton of an element D Σ on Ω r s ergodc provded that t has no roots of unty as egenvalues. In partcular, t follows that C Σ s ergodc. Now t follows from [2, Lemma II.15] that M conssts of torson elements. Suppose that M s nfnte. Then M M contans 0 as an accumulaton pont. Let y n Ω r be a sequence such that y n 0 and πy n M M. If h y n / V 1 := for nfntely many n, then we may argue exactly as n Case I of [3, p. 252] wth B = M. We conclude that M = Ω r, whch contradcts mnmalty of M. Hence, t remans to consder the case when every element x n a suffcently small neghbourhood of 0 n M M s of the form πy for some y V 1. We take an ergodc element D Σ and M M a D-mnmal subset. Then for every x M,we have D n sx x along a subsequence n k. In partcular, t follows that for some n N, j=1 V 1 j D n x x = πy 7 wth y V 1. It follows from 6 that there exsts an element E Σ such that E m y 0 asm. Passng to a subsequence, we also obtan E m s x z M. Hence, applyng E m s to both sdes of 7, we conclude that D n z = z, andby[2, Lemma II.15], z s a torson element. Snce M s Σ -mnmal, t must consst of torson elements. Proof of Proposton 3.1. We denote by Ω[l] the subset of elements whose order dvdes l. Wenote that Ω[l] s fnte see [2, Lemma II.13] and Σ-nvarant.

7 A. Gorodnk, S. Kadyrov / Journal of Number Theory Let M be a Σ-mnmal set contaned n a gven closed Σ-nvarant set. We use nducton on k. The case when k = 1 s handled by Lemma 3.2. In partcular, t follows that p 1 M contans a torson element of order l 1, where p 1 : Ω Ω r 1 a 1 denotes the projecton map. Let N = { y k =2 Ω r : x, y M for some x Ω r 1 a 1 [l 1 ] Snce N s non-empty, nvarant, and closed, t follows from the nductve hypothess that N contans a pont y such that l 2 y = 0 for some l 2 N. Then M contans x, y for some x Ω r 1 a 1 [l 1 ], and x, y Ω[l 1 l 2 ]. From Proposton 3.1, we also deduce. Lemma 3.3. Let M be a closed Σ-nvarant set. Then there exst s N and a torson pont r Msuchthat A s r = B s r = r. Proof. We recall that by [2, Lemma II.13] the set Ω[l], s fnte. Snce ths set s clearly Σ-nvarant, t follows from Proposton 3.1 that M contans a fnte Σ-nvarant set N consstng of torson elements. We pck N to be a mnmal set wth these propertes. Snce AN N s also Σ-nvarant, we conclude that AN = N and smlarly BN = N. Then t follows that A N and B N are bjectons of the fnte set N, and there exsts s N such that A N s = B N s = d, whch mples the lemma. 4. Approxmaton of long lne segments Let Υ denote the set of accumulaton ponts of Υ := πσ v = Σω. The am of ths secton s to show that one can approxmate projectons of arbtrary long lne segments by ponts n Υ.For ths we recall that Υ contans a torson element r see Proposton 3.1 and apply the acton of Σ to a sequence x s s 1 contaned n Υ and convergng to r. To produce nontrval lmts, one needs addtonal propertes of the sequence x s that are provded by the followng two lemmas. Lemma 4.1. For any pont x Υ there exsts a sequence x s Υ convergng to x such that where V 1 := k h =1 j=1 V 1. j x s = π y s + x wth y s / V 1, y s 0, Proof. To prove the lemma we use the assumpton that ξ / Qλ, μ for some = 1,...,k. Let x s s 1 be a sequence of dstnct ponts n Υ = πσω convergng to x. Wewrte x s = π y s + x, where y s s a sequence of ponts n Ω convergng to zero. More explctly, x s = π A ms B ns v = x s 1,...,xs for ms, ns N, where x s = π A ms B ns ξ v 11 = π λ ms μ ns ξ v 11. Recall that we have assumed that ξ / Qλ, μ for some = 1,...,k. We clam that for ths the sequence x s s 1 conssts of dstnct ponts. Indeed, suppose that x s 1 = x s 2 for some s 1 s 2.Then ms λ 1 μ ns 1 λ ms 2 μ ns 2 ξ v 11 kerπ. k

8 2506 A. Gorodnk, S. Kadyrov / Journal of Number Theory Snce the egenvector v 11 cannot be proportonal to a ratonal vector, we conclude that λ ms 1 μ ns 1 = λ ms 2 μ ns 2, and hence ms 1 = ms 2 and ns 1 = ns 2 because λ, μ s assumed to be multplcatvely ndependent. Then x s1 = x s2, whch gves a contradcton. Now f we suppose that y s satsfes y s V 1 for all suffcently large s, then we can apply the argument of Case II n [3, p. 253] to the sequence {x s. Ths argument yelds that ξ Qλ, μ, whch s a contradcton. Hence, by passng to a subsequence, we can arrange that y s / V 1, as requred. Gven a sequence y s s 1 as above, we denote by I the set of ndces, j,l such that y s jl 0. Lemma 4.2. In Lemma 4.1, we can pck a sequence y s s 1 so that for some D Σ, λ l D pj > 1 for all, j,l I, λ 1 l 1 D λ 2 l 2 D for all 1, j 1,l 1, 2, j 2,l 2 I wth 1,l 1 2,l 2. Proof. The proof reles on the ndependence property b of the man theorem of the pars λ, μ. We pck a sequence y s s 1 as n Lemma 4.1 wth a mnmal set of ndces I. Then by [2, Lemma II.7], foranyd Σ we have ether λ l D pj > 1forall, j,l I or λ l D pj 1forall, j,l I. Hence, t follows from the hyperbolcty assumpton c of the man theorem that ether A or B satsfes. Wthout loss of generalty, we assume that A satsfes. Then there exsts n 0 N such that A n B satsfes for all n n 0. Now we show that D := A n B for some n n 0 satsfes, whch s equvalent to showng that λ n a 1 μ a1 λ n a 2 μ a2 8 for all a 1 a 2 n the set J ={,l: 1 k, 1 l r. We say that a 1 a 2 f there exsts n N such that λ n a 1 = λ n a 2. It s easy to check that ths s an equvalence relaton and there exsts m 0 such that λ m 0 a 1 = λ m 0 a 2 for all a 1 and a 2 n the same equvalence class. It follows from the ndependence assumpton b of the man theorem and 4 that λ u a1,μ u a 1 λ u a2,μ u a 2 for all a 1 a 2 and u N. Thus, f a 1 and a 2 belong to the same equvalence class, then μ m 0 a 1 μ m 0 a 2 and, n partcular, μ a1 μ a2. Ths mples that 8 holds wthn the same equvalence class when n s a multple of m 0. Now we consder the case when a 1 a 2 belong to dfferent equvalence classes. If 8 falsforn and n,then λ n n a 1 = λ n n a 2, and n = n. Hence, n ths case 8 may fal only for fntely many n s. Hence, f we take n to be a suffcently large multple of m 0, then both and hold. We apply the argument of [2, Secton II.3] to the sequence y s s 1 and D Σ constructed n Lemma 4.2. Ths yelds the followng lemma cf. [2, Lemma II.11]. We say that a set Y s an ɛ-net for the set X f for every x X there exsts y Y wthn dstance ɛ from x.

9 A. Gorodnk, S. Kadyrov / Journal of Number Theory Lemma 4.3. Assume that Υ contans a torson pont r fxed by Σ. Then there exst D Σ, aprmep,j {, j,l I: p j = p, c b 0 wth b J n a fnte extenson of Q p,u Ω and t m satsfyng t m max λ b D m b J p when p =, max λ b D m b J p when p <, 9 p t m such that f we defne v m,t := D m u + t λ b D m c b v b, b J where t [0, t m ] when p =,andt p t mz p when p <, thenv m,t Ω and for every ɛ > 0 and m > mɛ,thesetπ 1 Υ r forms an ɛ-net for {v m,t. 5. Proof of the man theorem As n the prevous secton, Υ ={πa m B n v: m,n N, and Υ s the set of lmt ponts of Υ. We frst assume that Υ contans a torson pont r fxed by Σ and apply Lemma 4.3. Let λ := max λb D p and K := { b J : λb D p = λ. b J We take a sequence t m < t m such that when p =, and t m λm and t m max λb D m b J \K p 0 10 p t m λ m and p t m max λb D m b J \K p 0 11 when p <. Let w m,t = D m u + t λ b D m c b v b b K where t [0, t m ] when p =, and t pt m Zp when p <. It follows from 10 and 11 that for every ɛ > 0 and m > mɛ, {v m,t forms an ɛ-net for {w m,t. Ths shows that we may assume that n Lemma 4.3 λ b D p = λ for all b J.Wewrteλ b D = λω b where ω b p = 1. We clam that there exsts 1 m 0 J such that cm 0 := b J ω m 0 b c b Indeed, suppose that cm = 0forall1 m J. Ths mples that the J J -matrx λb D m b J, 1 m J

10 2508 A. Gorodnk, S. Kadyrov / Journal of Number Theory s degenerate. However, t follows from Lemma 4.2 that λ b1 D λ b2 D for b 1 b 2, whch s a contradcton. Hence, 12 holds. We clam that there exsts a subsequence m such that ω m ω m 0 for all b J.Toshow b b ths, we consder the rotaton on the compact abelan group { z p = 1 J defned by the vector ω b b J. Snce the orbt closure of the dentty s mnmal, t follows that ω m b b J 1,...,1 along a subsequence, and the clam follows. We consder the cases p = and p < separately. Suppose that p =. We observe that by 5, Π v m,t = z m + { tλm ω m b c b = z m + { tλ m cm b J mod 1, where z m = ΠD m u. Snce t m λ m and cm cm 0 0, we conclude that for all suffcently large, Π { v m,t 0 t t m = R/Z. On the other hand, for every ɛ > 0 and > ɛ, thesetπ 1 Υ r forms an ɛ-net for {v m,t 0 t tm. Therefore, snce Π s contnuous, t follows that ΠΥ r s dense n R/Z, whch completes the proof of the theorem. Now suppose that p <. In ths case, λ = p n, and Π v m,t = z m + { tp mn ω m b c b = p z m + { tp mn cm p b J mod 1. For all suffcently large, wehave cm p = cm 0 =p l.thus, { Π v m,t t p tm Z p = z m + { p t m nm +l Z p = z m + { 1 j=t m nm +l p c j p j :0 c j p 1 mod 1, and ths set s p t m nm +l -dense n R/Z. Sncep t m +nm,forallɛ > 0 and > ɛ ths set forms an ɛ-net for R/Z. On the other hand, for every ɛ > 0 and suffcently large, thesetπ 1 Υ r forms an ɛ-net for {v m,t t p tm Zp. Hence, we conclude that ΠΥ r s dense n R/Z. Ths completes the proof of the theorem under the assumpton that Υ contans a torson pont r fxed by Σ. To prove the theorem n general, we observe that by Lemma 3.3 there exst s N and a torson pont r Υ such that A s r = B s r = r. Then there exst 0 m 0,n 0 s 1 such that r s an accumulaton pont for {πa ms+m 0 B ns+n 0 v: m,n N. Applyng the above argument to the semgroup Σ = A s, B s and the vector v = A m 0 B m 0 v, we establsh the theorem n general. Acknowledgments The frst author s support by EPSRC, ERC and RCUK, and the second author s supported by EPSRC.

11 A. Gorodnk, S. Kadyrov / Journal of Number Theory References [1] D. Berend, Mnmal sets on tor, Ergodc Theory Dynam. Systems [2] D. Berend, Mult-nvarant sets on compact abelan groups, Trans. Amer. Math. Soc [3] D. Berend, Dense mod 1 dlated semgroups of algebrac numbers, J. Number Theory [4] B. Kra, A generalzaton of Furstenberg s Dophantne theorem, Proc. Amer. Math. Soc [5] R. Urban, On densty modulo 1 of some expressons contanng algebrac ntegers, Acta Arth [6] R. Urban, Algebrac numbers and densty modulo 1, J. Number Theory [7] R. Urban, Algebrac numbers and densty modulo 1, II, Unf. Dstrb. Theory