On a theorem of V. Bernik in the metrical theory of Diophantine approximation

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1 Acta Arthmetca 117, no.1 (2005) On a theorem of V. Bern n the metrcal theory of Dophantne approxmaton by V. Beresnevch (Mns) 1 1. Introducton. We begn by ntroducng some notaton: #S wll denote the number of elements n a fnte set S; the Lebesgue measure of a measurable set S R wll be denoted by S ; P n wll be the set of ntegral polynomals of degree n. Gven a polynomal P, H(P ) wll denote the heght of P,.e. the maxmum of the absolute values of ts coeffcents; P n (H) = {P P n : H(P ) = H}. The symbol of Vnogradov n the expresson A B means A CB, where C s a constant. The symbol means both and. Gven a pont x R and a set S R, dst(x, S) = nf{ x s : x S}. Throughout, Ψ wll be a postve functon. Mahler s problem. In 1932 K. Mahler [10] ntroduced a classfcaton of real numbers x nto the so-called classes of A, S, T and U numbers accordng to the behavor of w n (x) defned as the supremum of w > 0 for whch P (x) < H(P ) w holds for nfntely many P P n. By Mnovs s theorem on lnear forms, one readly shows that w n (x) n for all x R. Mahler [9] proved that for almost all x R (n the sense of Lebesgue measure) w n (x) 4n, thus almost all x R are n the S-class. Mahler has also conjectured that for almost all x R one has the equalty w n (x) = n. For about 30 years the progress n Mahler s problem was lmted to n = 2 and 3 and to partal results for n > 3. V. Sprndzu proved Mahler s conjecture n full (see [12]). A. Baer s conjecture. Let W n (Ψ) be the set of x R such that there are nfntely many P P n satsfyng (1) P (x) < Ψ(H(P )). 1 The wor has been supported by EPSRC grant GR/R90727/01 Key words and phrases: Dophantne approxmaton, Metrc theory of Dophantne approxmaton, the problem of Mahler 2000 Mathematcs Subject Classfcaton: 11J13, 11J83, 11K60 1

2 A. Baer [1] has mproved Sprndžu s theorem by showng that W n (Ψ) = 0 f Ψ 1/n (h) < and Ψ s monotonc. h=1 He has also conjectured a stronger statement proved by V. Bern [4] that W n (Ψ) = 0 f the sum (2) h n 1 Ψ(h) h=1 converges and Ψ s monotonc. Later V. Beresnevch [5] has shown that R \ W n (Ψ) = 0 f (2) dverges and Ψ s monotonc. We prove Theorem 1. Let Ψ : R R + be arbtrary functon (not necessarly monotonc) such that the sum (2) converges. Then W n (Ψ) = 0. Theorem 1 s no longer mprovable as, by [5], the convergence of (2) s crucal. Notce that for n = 1 the theorem s smple and nown (see, for example, [8, p. 121]). Therefore, from now on we assume that n Subcases of Theorem 1. Let δ > 0. We defne the followng 3 sets denoted by W bg (Ψ), W med (Ψ) and W small (Ψ) consstng of x R such that there are nfntely many P P n smultaneously satsfyng (1) and one of the followng nequaltes (3) 1 P (x), (4) H(P ) δ P (x) < 1, (5) P (x) < H(P ) δ respectvely. Obvously W n (Ψ) = W bg (Ψ) W med (Ψ) W small (Ψ). Hence to prove Theorem 1 t suffces to show that each of the sets has zero measure. Snce sum (2) converges, H n 1 Ψ(H) tends to 0 as H. Therefore, (6) Ψ(H) = o(h n+1 ) as H. 2

3 3. The case of a bg dervatve. The am of ths secton s to prove that W bg (Ψ) = 0. Let B n (H) be the set of x R such that there exsts a polynomal P P n (H) satsfyng (3). Then (7) W bg (Ψ) = N=1 H=N B n (H). Now W bg (Ψ) = 0 f W bg (Ψ) I = 0 for any open nterval I R satsfyng (8) 0 < c 0 (I) = nf{ x : x I} < sup{ x : x I} = c 1 (I) <. Therefore we can fx an nterval I satsfyng (8). By (7) and the Borel-Cantell Lemma, W bg (Ψ) I = 0 whenever (9) B n (H) I <. H=1 By the convergence of (2), condton (9) wll follow on showng that (10) B n (H) I H n 1 Ψ(H) wth the mplct constant n (10) ndependent of H. Gven a P P n (H), let σ(p ) be the set of x I satsfyng (3). Then (11) B n (H) I = σ(p ). P P n (H) Lemma 1. Let I be an nterval wth endponts a and b. Defne the followng sets I = [a, a + 4Ψ(H)] [b 4Ψ(H), b] and I = I \ I. Then for all suffcently large H for any P P n (H) such that σ(p ) I, for any x 0 σ(p ) I there exsts α I such that P (α) = 0, P (α) > P (x 0 ) /2 and x 0 α < 2Ψ(H) P (α) 1. The proof of ths Lemma nearly concdes wth the one of Lemma 1 n [5] and s left for the reader. There wll be some changes to constants and notaton and one also wll have to use (6). Gven a polynomal P P n (H) and a real number α such that P (α) 0, defne σ(p ; α) = {x I : x α < 2Ψ(H) P (α) 1 }. Let I and I be defned as n Lemma 1. For every polynomal P P n (H), we defne the set Z I (P ) = {α I : P (α) = 0 and P (α) 1/2}. 3

4 By Lemma 1, for any P P n (H) we have the ncluson (12) σ(p ) I σ(p ; α). α Z I (P ) Gven Z wth 0 n, defne P n (H, ) = {P = a n x n + + a 0 P n (H) : a = 0} and for R P n (H, ) let P n (H,, R) = {P P n (H) : P R = a x }. It s easly observed that (13) P n (H) = n =0 R P n (H,) P n(h,, R) and (14) #P n (H, ) H n 1 for every. Tang nto account (11), (13), (14) and that I Ψ(H), t now becomes clear that to prove (10) t s suffcent to show that for every fxed and fxed R P n (H, ) (15) σ(p ) I Ψ(H). P P n (H,,R) Let and R be fxed. Defne the ratonal functon R(x) = x R(x). By (8), there exsts a collecton of ntervals [w 1, w ) I ( = 1,..., s), whch do not ntersect parwse and cover I, such that R(x) s monotonc and does not change the sgn on every nterval [w 1, w ). It s clear that s depends on n only. Let Z I,R = P P Z n(h,,r) I(P ), = #(Z I,R [w 1, w )) and Z I,R [w 1, w ) = {α (1),..., α ( ) }, where α (j) < α (j+1). Gven a P P n (H,, R), we obvously have the dentty x P (x) x 1 P (x) x 2 = ( ) P (x) = R(x). Tang x to be α Z I (P ) leads to P (α) α = R(α). By (8), P (α) R(α). Now, by Lemma 1, σ(p ; α) Ψ(H) P (α) 1 Ψ(H) R (α) 1. x 4

5 Usng (12), we get P P n (H,,R) σ(p ) I s Ψ(H) =1 j=1 1 (j) R(α ) Now to show (15) t suffces to prove that for every (1 s) (16) j=1 R (α (j) ) 1 1. Fx an ndex (1 s). If 2 then we can consder two sequental roots α (j) and α (j+1) of two ratonal functons R + a,j and R + a,j+1 respectvely. For convenence let us assume that R s ncreasng and postve on [w 1, w ). Then R s strctly monotonc on [w 1, w ), and we have a,j a,j+1. It follows that a,j a,j+1 1. Usng The Mean Value Theorem and the monotoncty of R, we get where α (j) α (j+1) 1 j=1 1 a,j 0 a,j+1 0 = R (α (j) ) R (α (j+1) ) = = R ( α (j) ) α (j) α (j) α (j+1) s a pont between α (j), whence we readly get R (α (j+1) ) 1 1 j=1 ( α (j+1) R (α (j+1) ) α (j) α (j+1), and α (j+1). Ths mples R (α (j+1) ) 1 ) α (j) = α ( ) α (1) w w 1. The last nequalty and R (α (1) ) P (α (1) ) 1 yeld (16). It s easly verfed that (16) holds for every wth 2 and s certanly true when = 1 or = 0. Ths completes the proof of the case of a bg dervatve. 4. The case of a medum dervatve. As above we fx an nterval I satsfyng (8). The statement W med (Ψ) = 0 wll now follow from W med (Ψ) I = 0. We wll use the followng Lemma 2 (see Lemma 2 n [6]). Let α 0,..., α 1, β 1,..., β R {+ } be such that α 0 > 0, α j > β j 0 for j = 1,..., 1 and 0 < β < +. 5

6 Let f : (a, b) R be a C () functon such that nf x (a,b) f () (x) β. Then, the set of x (a, b) satsfyng { f(x) α 0, β j f (j) (x) α j (j = 1,..., 1) s a unon of at most ( + 1)/2 + 1 ntervals wth lengths at most mn 06<j6 3 (j +1)/2 (α /β j ) 1/(j ). Here we adopt c = + for c > 0. 0 Gven a polynomal P P n (H), we redefne σ(p ) to be the set of solutons of (4). Snce P (n) (x) = n!a n, we can apply Lemma 2 to P wth = n and α 0 = Ψ(H), α 1 = 1, β 1 = nf P (x) H δ, β n = 1, x σ(p ) α 2 = = α n 1 = +, β 2 = = β n 1 = 0. Then we conclude that σ(p ) s a unon of at most n(n + 1)/2 + 1 ntervals of length α 0 /β 1. There s no loss of generalty n assumng that the sets σ(p ) are ntervals as, otherwse, we would treat the ntervals of σ(p ) separately. We also can gnore those P for whch σ(p ) s empty. For every P we also defne a pont γ P σ(p ) such that nf x σ(p ) P (x) 1 P (γ 2 P ). The exstence s easly seen. Now we have (17) σ(p ) Ψ(H) P (γ P ) 1. It also follows from the choce of γ P that (18) H(P ) δ P (γ P ) < 1. Now defne expansons of σ(p ) as follows: σ 1 (P ) := {x I : dst(x, σ(p )) < (H P (γ P ) ) 1 }, σ 2 (P ) := {x I : dst(x, σ(p )) < H 1+2δ }. By (4), σ 1 (P ) σ 2 (P ). Moreover, t s easy to see that (19) σ 1 (P ) σ 2 (Q) for any Q P n (H) wth σ 1 (Q) σ 1 (P ). It s readly verfed that σ 1 (P ) (H P (γ P ) ) 1, and therefore, by (17), σ(p ) σ 1 (P ) HΨ(H). 6

7 Tae any x σ 2 (P ). Usng the Mean Value Theorem, (18) and x γ P H 1+2δ, we get P (x) P (γ P ) + P ( x)((x γ P ) 1 + H H 1+2δ H 2δ, where x s between x and γ P. Smlarly we estmate P (x) resultng n (20) P (x) H 1+4δ, P (x) H 2δ for any x σ 2 (P ). Now for every par (, m) of ntegers wth 0 < m n we defne P n (H,, m) = {R = a n x n + + a 0 P n (H) : a = a m = 0} and for a gven polynomal R P n (H,, m) we defne P n (H,, m, R) = {P = R + a m x m + a x P n (H)}. The ntervals σ(p ) wll be dvded nto 2 classes of essental and nonessental ntervals. The nterval σ(p ) wll be essental f for any choce of (, m, R) such that P P n (H,, m, R) for any Q P n (H,, m, R) other than P we have σ 1 (P ) σ 1 (Q) =. For fxed, m and R summng the measures of essental ntervals gves σ(p ) HΨ(H) σ1 (P ) HΨ(H) I HΨ(H). As #P n (H,, m) H n 2 and there are only n(n + 1)/2 dfferent pars (, m) we obtan the followng estmate σ(p ) H n 1 Ψ(H). essental ntervals σ(p ) wth P P n(h) Thus, by the Borel-Cantell Lemma and the convergence of (2), the set of ponts x of W med (Ψ) I whch belong to nfntely many essental ntervals s of measure zero. Now let σ(p ) be non-essental. Then, by defnton and (19) there s a choce of, m, R such that P P n (H,, m, R) and there s a Q P n (H,, m, R) dfferent from P such that σ(p ) σ 1 (P ) σ 2 (P ) σ 2 (Q). On the set σ 2 (P ) σ 2 (Q) both P and Q satsfy (20) and so does the dfference P (x) Q(x) = b m x m + b x. It s not dffcult to see that b m 0 f H s bg enough. Therefore usng (20)we get (21) x m + b H 1+4δ H 1+4δ and max{ b m, b } H 2δ. b m b m 7

8 Now let x belongs to nfntely many non-essental ntervals. Wthout loss of generalty we assume that x s transcendental as otherwse t belongs to a countable set, whch s of measure zero. Therefore (21) s satsfed for nfntely many b m, b Z. Hence, the nequalty x m p < q 1 5δ 2δ q holds for nfntely many p, q Z. Tang δ = 1 1 5δ so that becomes 2 + δ 10 2δ and applyng standard Borel-Cantell arguments (see [8, p. 121]) we complete the proof of the case of a medum dervatve for non-essental ntervals. 5. The case of a small dervatve. In ths secton we prove that W small (Ψ) = 0. We wll mae use of Theorem 1.4 n [7]. By tang d = 1, f = (x, x 2,..., x n ), U = R, T 1 =... = T n = H, θ = H n+1, K = H δ n that theorem, we arrve at mn(δ,n 1) Theorem 2. Let x 0 R and δ =. Then there exsts a fnte (n+1)(2n 1) nterval I 0 R contanng x 0 and a constant E > 0 such that { } x I 0 : P (x) < H n+1, P (x) < H δ EH δ. P P n, 0<H(P )6H In partcular Theorem 2 mples that for any δ > 0 the set of x R, for whch there are nfntely many polynomals P P n satsfyng the system (22) P (x) < H(P ) n+1, P (x) < H(P ) δ, has zero measure. Indeed, ths set conssts of ponts x I 0 whch belong to nfntely many sets τ m = {x I 0 : (22) holds for some P P n wth 2 m 1 < H(P ) 2 m } By Theorem 2, τ m 2 mδ wth δ > 0. Therefore, m=1 τ m < and the Borel-Cantell Lemma completes the proof of the clam. Tang nto account (6), ths completes the proof of the case of a small dervatve and the proof of Theorem Concludng remars. An analogue of Theorem 1 when P s assumed to be rreducble over Q and prmtve (.e. wth coprme coeffcents) can also be sought. To mae t more precse, let P n(h) be the subset of 8

9 P n (H) consstng of prmtve rreducble polynomals P of degree deg P = n and heght H(P ) = H. Now the set of prmtve rreducble polynomals of degree n s Pn = H=1 P n(h). Let Wn(Ψ) be the set of x R such that there are nfntely many P P satsfyng (1). Theorem 3. Let Ψ : R R + be arbtrary functon such that the sum (23) H=1 converges. Then W n(ψ) = 0. #P n(h) H Ψ(H) For n = 1 the proof of Theorem 3 s a straghtforward applcaton of the Borel-Cantell Lemma and we agan refer to [8, p. 121]. For n > 1 the proof follows from the followng 2 observatons: 1) W n(ψ) W n (Ψ) and 2) #P n(h) H n. The second one guarantees the converges of (2), whch now mples 0 W n(ψ) W n (Ψ) = 0. The proof of the relaton #P n(h) H n s elementary and s left for the reader. In fact, one can easly estmate the number of prmtve reducble polynomals n P n and tae them off the set of all prmtve polynomals n P n whch s well nown to contan at least constant #P n (H). The Duffn-Schaeffer conjecture. The conjecture states that for n = 1 f (23) dverges then R \ W n(ψ) = 0. The multple #P 1 (H) n sum (23) becomes ϕ(h), where ϕ s the Euler functon. The followng problem can be regarded as the generalzaton of the Duffn-Schaeffer conjecture for ntegral polynomals of hgher degree: Prove that R \ W n(ψ) = 0 whenever (23) dverges. Alternatvely, for n > 1 one mght nvestgate the measure of R\W n (Ψ). So far t s unclear f for n > 1 R\W n(ψ) = 0 s equvalent to R\W n (Ψ) = 0, whch s another ntrcate queston. A remar on manfolds. In the metrc theory of Dophantne approxmaton on manfolds one usually studes sets of Ψ-approxmable ponts lyng on a manfold wth respect to the measure nduced on that manfold. Mahler s problem and ts generalsatons can be regarded as Dophantne approxmaton on the Veronese curve (x, x 2,..., x n ). A pont f R n s called Ψ-approxmable f a f < Ψ( a ), 9

10 for nfntely many a Z n, where a = max 166n a for a = (a 1,..., a n ), x = mn{ x z : z Z} and Ψ : R R +. Let f : U R n be a map defned on an open set U R d. We say that f s non-degenerate at x 0 U f for some l N the map f s l tmes contnuously dfferentable on a suffcently small ball centered at x 0 and there are n lnearly ndependent over R partal dervatves of f at x 0 of orders up to l. We say that f s non-degenerate f t s non-degenerate almost everywhere on U. The non-degeneracy of a manfold s naturally defned va the non-degeneracy of ts local parametersaton. In 1998 D. Klenboc and D. Marguls proved the Baer-Sprndžu conjecture by showng that any non-degenerate manfold s strongly extremal. In partcular, ths mples an analogue of Mahler s problem for non-degenerate manfolds. A few years later an analogue of A. Baer s conjecture wth monotonc Ψ (normally called a Groshev type theorem for convergence) has ndependently been proven by V. Beresnevch [6] and by V. Bern, D. Klenboc and G. Marguls [7] for non-degenerate manfolds. It s also remarable that the proofs were gven wth dfferent methods. The dvergence counterpart (also for monotonc Ψ) has been establshed n [3]. In [7] a multplcatve verson of the Groshev type theorem for convergence has also been gven. Theorem 1 of ths paper can be readly generalsed for non-degenerate curves: Gven a non-degenerate map f : I R n defed on an nterval I, for any functon Ψ : R R + such that the sum (2) converges for almost all x I the pont f(x) s not Ψ-approxmable. Even further, usng the slcng technque of Pyartly [11] one can extend ths for a class of n-dfferentable non-degenerate manfolds whch can be folated by non-degenerate curves. In partcular, ths class ncludes arbtrary non-degenerate analytc manfold. However wth the technque n our dsposal we are currently unable to prove the followng Conjecture. Let f : U R n be a non-degenerate map, where U s an open subset of R d. Then for any functon Ψ : R R + such that the sum (2) converges for almost all x U the pont f(x) s not Ψ-approxmable. References [1] A. Baer, On a theorem of Sprndžu, Proc. Royal Soc. Seres A 292 (1966),

11 [2], A concse ntroducton to number theory, Cambrdge Unversty Press, Cambrdge, [3] V.V. Beresnevch, V.I. Bern, D.Y. Klenboc, and G.A. Marguls, Metrc Dophantne approxmaton: The Khntchne Groshev theorem for non-degenerate manfolds, Moscow Mathematcal Journal 2 (2002), no. 2, [4] V.I. Bern, On the exact order of approxmaton of zero by values of ntegral polynomals, Acta Arthmetca 53 (1989), 17 28, (In Russan). [5] V.V. Beresnevch, On approxmaton of real numbers by real algebrac numbers, Acta Arthmetca 90 (1999), no. 2, [6], A Groshev type theorem for convergence on manfolds, Acta Mathematca Hungarca 94 (2002), no. 1-2, [7] V.I. Bern, D.Y. Klenboc, and G.A. Marguls, Khntchne-type theorems on manfolds: the convergence case for standard and multplcatve versons, Internatonal Mathematcs Research Notces (2001), no. 9, [8] J.W.S. Cassels, An ntroducton to Dophantne Approxmaton, Cambrdge Unversty Press, Cambrdge, [9] K. Mahler, Über das Maßder Menge aller S-Zahlen, Math. Ann. 106 (1932), [10], Zur Approxmaton der Exponental functon und des Logarthmus, J. rene und angew. Math. 166 (1932), [11] A. Pyartl, Dophantne approxmaton on submanfolds of eucldean space, Funts. Anal. Prlosz. 3 (1969), 59 62, (In Russan). [12] V.G. Sprndžu, Mahler s problem n the metrc theory of numbers, vol. 25, Amer. Math. Soc., Provdence, RI, 1969, Translatons of Mathematcal Monographs. Insttute of Mathematcs Academy of Scences of Belarus , Surganova 11, Mns, Belarus beresnevch@m.bas-net.by 11