CONSTRUCTION OF A NUMBER GREATER THAN ONE WHOSE POWERS ARE UNIFORMLY DISTRIBUTED MODULO ONE. Miguel A. Lerma. February 5, 1996

Transcription

1 COSTRUCTIO OF A UMBER GREATER TA OE WOSE POWERS ARE UIFORMLY DISTRIBUTED MODULO OE Miguel A. Lerm Februry 5, 996 Abstrct. We study how to construct number greter thn one whose powers re uniformly distributed modulo. Also we prove tht for every λ > 0 there is dense set of computble numbers α > such tht the discrepncy of {λα n } n= is O( 4+ε.. Introduction. It is well known tht, s consequence of Koksm s Theorem [], for lmost every number α > (in the sense of Lebesgue mesure, the sequence {α n } n= is uniformly distributed modulo (u.d. mod, but only exmples from the exceptionl set, such s P.V.-numbers nd Slem numbers, re known [, pg. 7]. Another well known metric result is Weyl s theorem: given ny α >, the sequence {λα n } n= is u.d. mod. for lmost every rel number λ [3]. owever, in this cse explicit exmples of such λ re known. When α is n integer, the problem reduces to the construction of norml number to the bse α [4, chp., secc. 8]. If α is not n integer, the construction of λ is somewht more complicted, but still possible, s proven by Kulikov [5], by using n ide of Lebesgue permitting the effective use of metricl theorems [6]. It is interesting to note tht some purely existentil results cn be trnsformed into constructive procedures suitble to produce mthemticl objects with the required property. For instnce, Gry [7] hs used Cntor s result to design n lgorithm which genertes the digits of trnscendentl number η in the intervl (0,. Bsiclly, his lgorithm genertes (suitble deciml pproximtions of ll lgebric numbers by orderly generting ll polynomils of integrl coefficients, nd pproximting its roots up to some point. Then the digits of η re defined by digonl method. Similrly, Kulikov s result lso tkes dvntge of metricl result to give procedure which genertes the digits of number with the required property [5]. A similr ide will be used here. Typeset by AMS-TEX

2 MIGUEL A. LERMA. Previous results. Given λ > 0, we re going to define procedure to construct number α 0 > such tht the discrepncy of {λα n 0 } n= pproches zero s. The ide is to strt with some closed intervl I = [, b], with < < b, nd tke open subsets J, J,... of points α for which {λα n } n= hs high discrepncy. Those subsets will hve the following fetures: (i Ech J k is union of finitely mny open intervls with computble endpoints. (ii G = I \ k= J k, nd for every α G the discrepncy of {λα n } n= tends to zero s. (iii There is computble sequence of nested closed intervls I i of length pproching zero such tht I i \ k= J k for every i. The number α 0 will be determined s the intersection of the I i s. In order to control the discrepncy we re going to use the following bound [8]: ( (x,..., x + + h= { πh + } + n= e πihx n where (x,..., x is the discrepncy of x,..., x. The following lemm gives bounds for the size of the subsets of I with high discrepncy. Lemm. Let λ be ny fix positive rel number. Let I = [, b] ny closed intervl of rel numbers such tht < < b. For positive integers nd, let φ, (α be the function: φ, (α = Also define the set: + + E(,, τ = h= { πh + } + { α I : + log nd let 0 be fix integer such tht 0 (i For 0 the following inequlity holds: µ(e(,, τ < (b + log where µ represents the Lebesgue mesure. e πihλαn n= } < φ, (α, λ( (b. Then: { ( + log π (ii Let ν be ny rel number greter tht. Let E M be the set: E M = E([M ν ], [M ], + } +

3 COSTRUCTIO OF A UMBER GREATER TA OE... 3 where [x] = integer prt of x. Then, for M 0 : µ(e M < (b M ν (iii Given ny pir of rel numbers ε > ε > 0, ssume ν = + ε /. Let J(M, M be the (possibly empty set J(M, M = M M=M + E M where M M 0 (M my be. Then µ(j(m, M < 4 (b ε M ε / For ny integer M 0 > mx {(4/ε /ε, }, the set G M0 = I \ J(M 0, hs positive mesure. If α G M0 then the discrepncy of {λα n } n= verifies ( log = O = O( 4+ε 4+ε 0 Proof. (i In the inequlity defining E(,, τ integrte the left hnd side over E(,, τ nd the right hnd side over the whole intervl [, b]: + log µ(e(,, τ < (b + + h= { πh + } + e πihλαn dα n= By Jensen s inequlity: ( b e πihλαn dα (b n= (b e πihλαn dα n= The integrl on the right hnd side cn be bounded in the following wy: e πihλαn b ( dα = n= n,m = (b + e πihλ(αm α n dα n<m e πihλ(αm α n dα

4 4 MIGUEL A. LERMA We hve [4, lemm..]: From here we get: e πihλ(αm α n dα < hλ(m m n n e πihλαn dα < (b + n= (b + n<m hλ( hλ(m m n n Going bck we get: For (b e πihλαn dα < ( + n= λ( (b (b we get: ( + / hλ( (b e πihλαn dα < n= hλ( (b ence: + log µ(e(,, τ < (b + < (b + h= { πh + } + (b { ( + log π + + } From here the nnounced result follows. (ii From (i we get: { ( } µ(e M < (b M M ν + 4 [M ] π + (b { M ν + 4M ν ( } [M ν ] π + { ( } (b M ν + 8 π + < (b M ν

5 COSTRUCTIO OF A UMBER GREATER TA OE... 5 (iii We hve: µ(j(m, M M M=M + < (b < (b = 4 (b ε M ε / µ(e M M=M + M dx x ν M ν Using the condition on M 0, we get tht J(M, M hs mesure less thn (b, hence G M0, which is its complement respect to I = [, b], hs positive mesure. Concerning the order of, we strt noting tht α G M0 implies φ, (α + log ( log = O / 4+ε for = [M ν ] nd = [M ]. ence, it remins only to prove the result for other vlues of. Assume tht [M ] < < [(M + ], nd = [M ν ]. For simplicity, put = [M ] nd = [(M + ]. Then: φ, (α φ, (α { πh + } + h= < φ, (α + ( + log + + π ( log ( = O + O + O(log 4+ε 4+ε Finlly we hve: < [(M + ] [M ] [M ] nd from here the nnounced result follows. n= + = O(M = O( 4+ε e πihλαn 3. The procedure. Given ny closed intervl of rel numbers I = [, b] with < < b, ny λ > 0, nd ny ε > 0, we re going to get number α 0 such tht α 0 I, nd the discrepncy of {λα0 n } n= is O( 4+ε, hence {λα n 0 } n= is u.d. mod.

6 6 MIGUEL A. LERMA Tke ν = + ε, where ε > ε > 0. Let E M subsets of I = [, b] s in the lemm. We define sequence of open sets {J k } k=, where J k is of the form J k = J(M k, M k, nd sequence of nested closed intervls {I k } k=0, where I k = [ k, b k ], in recursive wy. In step 0, we tke ny M 0 > mx {(4/ε /ε, 0 }, s in prt (iii of the lemm, nd I 0 = [ 0, b 0 ] equl to I = [, b]. ence we hve µ(i 0 \ J(M 0, > d 0 = (b 0 0 4(b /(ε M ε / 0 > 0 In step k, ssume tht I k = [ k, b k ] nd J i for i =,..., k hve lredy been found, nd tht µ(i k \ J(M 0, > d k > 0. ext, we tke M k M k such tht d k / > 4(b /(ε M ε / k. ow we form the set J k = J(M k, M k, dd it to the previously found to form the set J(M 0, M k = k i= J i, nd determine the mesure of I k \ J(M 0, M k for I k equl to ech one of I k = [ k, k +b k ] nd I k = [ k +b k, b k ]. Since µ(i k \ J(M 0, M k + µ(i k \ J(M 0, M k = µ(i k \ J(M 0, M k > µ(i k \ J(M 0, > d k t lest one of I k \ J(M 0, M k or I k \ J(M 0, M k should hve mesure greter thn d k /. So, we tke I k with the condition µ(i k \ J(M 0, M k > d k /. If we now subtrct 4(b /(ε M ε / k, which by prt (iii of the lemm is n upper bound for µ(j(m k,, we get positive lower bound d k = d k / 4(b /(ε M ε / k > 0 for µ(i k \ J(M 0,. At this point we hve found I k nd J k, nd we hve tht µ(i k \ J(M 0, > d k > 0, so everything is redy to proceed with step k +. Let α 0 be the unique point in k=0 I k. If G M0 is the closed set I \ J(M 0,, we hve I k G M0 = I k \ J(M 0, for every k, hence α 0 G M0, nd by prt (iii of the lemm, α 0 verifies the requirements. 4. Computtionl considertions nd summry. The clim tht α 0 is computble number rests on the the fct tht the sets E M re finite unions of open intervls with computble endpoints, nd so re the sets J k. To be more precise, the endpoints re solutions in α of equtions of the form: ( + + h= { πh + } + n= e πihλαn + log = 0 In prctice it is impossible to compute exctly such solutions, so it is necessry to del with pproximtions. Thn cn be done in such wy tht the pproximtions verify + log φ, (α + log for some fix τ τ close to τ, sy τ = τ ε 0. The sets E M computed this wy will be slightly smller thn the sets E M used bove, so the process cn still be crried

7 COSTRUCTIO OF A UMBER GREATER TA OE... 7 out successfully nd n α 0 I be found. owever the speed of convergence of the discrepncy of {λα n 0 } n= will be relxed to: But the result is still of the form if 0 < ε 0 < 4+ε 4+ε. ( log = O 4+ε ε 0 = O( 4+ε The procedure described here could be unprcticl becuse it would require very high computtionl lod. owever it does provide result of t lest theoreticl interest: there is dense set of computble numbers α > such tht {λα n } n= is u.d. mod. nd the discrepncy of {λα n } n= is O( 4+ε. References. J.F. Koksm, Ein mengentheoretischer Stz über die Gleichverteilung modulo Eins, Composition Mthemtic (953, M.J. Bertin et l, Pisot nd Slem umbers, Birkhäuser Verlg, Berlin, Weyl, Über die Gleichverteilung von Zhlen modulo Eins, Mth. Ann. 77 (96, L. Kuipers &. iederreiter, Uniform Distribution of Sequences, John Wiley & Sons, ew York, M.F. Kulikov, A construction problem concerned with the distribution of the frctionl prts of n exponentil function, Dokl. Akd. uk SSSR 43 (96, 5 54 (Russin; English trnsl. in Soviet Mth. Dokl. 3 (96, M.. Lebesgue, Sur certines démonstrtions d existence, Bull. Soc. Mth. Frnce 45 (97, Robert Gry, George Cntor nd Trnscendentl umbers, Amer. Mth. Moth. 0 (994, Jeffrey D. Vler, Some Extreml Functions in Fourier Anlysis, Bull. Amer. Mth. Soc. (ew Series (985, 83 6.