DIOPHANTINE APPROXIMATION WITH FOUR SQUARES AND ONE K-TH POWER OF PRIMES

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1 oural of atatical Scics: Advacs ad Alicatios Volu 6 Nubr 00 Pas -6 DOPHANNE APPROAON WH FOUR SQUARES AND ONE -H POWER OF PRES Dartt of atatics ad foratio Scic Ha Uivrsit of Ecooics ad Law Zzou 000 P. R. Cia -ail: wli7@aoo.co.c Scool of atatics ad foratio Scic Nort Cia Uivrsit of Watr Cosrvac ad Elctric Powr Zzou 00 P. R. Cia Abstract W sow tat if ar o-zro ral ubrs ot all of t sa si is ral ad at last o of t ratios i is irratioal t for 0 < σ < ad a ositiv itr t iqualit σ < ( a ) as ifiitl a ris solutios ( ).. troductio 96 Davort ad Hilbro [] rovd tat if ar o-zro ral ubrs ot all of t sa si wit o at last of t 00 atatics Subct Classificatio: D7 P. words ad rass: dioati aroiatio ri Davort-Hilbro tod. Rcivd Auust 00; Rvisd Stbr Scitific Advacs Publisrs

2 ratios irratioal t for a > 0 tr ist itrs i ot all zro suc tat <. (.) Latr irc ad Davort [] sowd tat a solutio of (.) ists wit all of ordr for a > 0 tus tr ist ifiitl a sts of itrs satisfi t iqualit i < a. Furtr aba [] tdd Davort ad Hilbro s rsult [] ad rovd if ar o-zro ral ubrs ot all of t sa si ad at last o ratio is irratioal t for a ositiv i itr a ral ubr ad a > 0 t iqualit < (.) as ifiitl a solutios i ositiv itrs. 978 Lau ad Liu [6] av a quatitativ for of aba s rsult. Our uros r is to ivstiat tat udr so cssar coditios wtr t rsult of aba [] is tru w itrs i (.) ar rlacd b ris. rsult w obtai is as follows: or. Suos tat ar o-zro ral ubrs ot all of t sa si tat is ral ad tat at last o of t ratios i is irratioal t for t iqualit 0 < σ < ad a ositiv itr < ( a ) (.) as ifiitl a ris solutios ( ). σ

3 DOPHANNE APPROAON WH FOUR SQUARES W ot t raso w lt itr i our tor is tat for t cas (.) is trivial sic t autors [7] av obtaid t aalou rsult about dioati aroiatio wit o ri ad tr squars of ris ad for t cas o a s t rsults of Vaua [9] ad Hara [].. Notatio rouout itr ad ar sall ositiv ubrs. Costats bot licit ad ilicit i t O ad otatios sall dd ol o ad. Witout loss of ralit w a assu tat is irratioal t tr ar ifiitl a airs of itrs q a wit a q ( a q) q > 0 ad a 0. W coos q to b lar i trs of ad. W writ ( πi) ad a t followi dfiitios. L q L lo τ 6 (.) q 6 Q ( ) ω 0 < ω < 6 ( ) lo ( ) lo. (.) Lt furtr si π ( 0) ( 0) π t i( ). (.)

4 . Outli of t tod Our tod os bac to Vaua [8] [9] ad t basis of t Davort-Hilbro adatio of t circl tod is t idtit 0 a (.) wic is a trivial corollar of La of Davort ad Hilbro []. La.. Lt γ β ρ i b t zros of t Ria Zta fuctio. Suos tat Y t. lo lo lo ρ ρ β γ Y Y Y O Y Y Y Proof. S [8]. W ot tat i d d. i Usi La. ad artial itratio w av ρ ρ β γ lo lo Y Y Y O Y Y d Y ρ ρ β γ Y d Y dy Y :

5 DOPHANNE APPROAON WH FOUR SQUARES R Y d O Y d ρ β γ L d O d ρ β γ L O. : (.) Ad so d ρ β γ L O. : (.) O itrcai t ordr of suatio ad itratio ad b (.) w av

6 6 lo 0 a lo 6 < L 6 : L c. 6 L (.) Nt w sall rov tat t itral i (.) is i ordr of aitud. As usual w slit t ra of ifiit itratio ito tr sctios { } { } { } : : : 6 6 > < t τ τ wic ar traditioall ad t iborood of t orii t itrdiat rio ad t trivial rio. doiat cotributio to tis ifiit itral is fro t iborood of t orii ad tr is a ood aroiatio to. cotributio fro t trivial rio is liibl. o dal wit t cotributio fro t

7 DOPHANNE APPROAON WH FOUR SQUARES 7 itrdiat rio w us t fact tat is irratioal wic abls us to sow o of ad ust b rlativl sall i t itrdiat rio.. Niborood of t Orii is art of t itral fors t ai ositiv cotributio to t itral. La 8 of Vaua [9] w t Las. ad.. La.. For w av i ( ) τ τ L τ L. τ La.. W av ( ) ( ) i ( ) τ L τ L. τ τ La.. W av L.

8 8 Proof. (.) (.) (.) Las. ad. w av ( ) i i i i > < ( ) ( ) ( ) ( ) ( ). (.) Usi Scwarz's iqualit w t ( ) ( ). L (.) Siilarl ( ) L (.)

9 DOPHANNE APPROAON WH FOUR SQUARES 9 ( ) L (.) ( ) L (.) ( ). L (.6) t dsird boud ow follows fro (.)-(.6). La.. W av > τ ad. Proof. Las. ad. w av. > τ τ Fro (.) w t 0 a d d 0 a 6 d. d (.7)

10 0 Lt ad sic ar ot all t sa si w a assu tat 0 0 < > ad t otr cass ca b dalt wit b t sa tod t. asd o > < w ta 8 a < trfor (.7) 8 6 wic ilis t clai.

11 DOPHANNE APPROAON WH FOUR SQUARES. trdiat Rio followi La. las a iortat rol i t tratt of t itrdiat rio. La.. Suos tat α as a ratioal aroiatio b ( b r) ad α < t iv a ral ubr ω 0 r > r ω Λ α r r wr t ilicit costat dds ol o ω. b r wit Proof. is is actuall or of Gos []. La.. For vr ral ubr lt V i ( S S ) t ω V. Proof. For coos a q suc tat wit ( a ) ad q Q. t q a q ( q Q ) (.) (.) ad (.) τ > Q c a a 0. f q q 6 a q aq a q qq a q a q a 6 qq Q < q. q

12 W rcall tat q was cos as t doiator of a covrt to t cotiud fractio for. us b Ldr s law of bst aroiatio w av q a > q for all itrs (.) a q wit q < q trfor aq q. Howvr b aq < 6 qq tis is a cotradictio tus w stablis tat for at last o 6 < q Q. La. ad wit α b a r q iv t dsird iqualit. La.. W av L. Proof. (.) ad Hua s iqualit w av [ ] 0 [ ] L (.)

13 DOPHANNE APPROAON WH FOUR SQUARES [ ] [ ] 0 ω (.) wr ω is a iv vr sall ositiv ubr. Dfi a V V b (.) (.) ad Höldr s iqualit w av V V V V

14 L ω ω. L is rovs t la. 6. rivial Rio La 6.. W av. L t Proof. Scwarz s iqualit ad Parsval s idtit w av t t 6 lo d 6 lo > d 6 > L

15 DOPHANNE APPROAON WH FOUR SQUARES L. 7. Proof of or Las... ad 6. totr il tat ( ) so b (.) w ow ( ) L i.. tr ar L ordrd ris ( ) wit ad <. Fro (.) ad (.) w σ L ifiit a ordrd ris ( ). ow ( a ) ad is lar sur tat (.) occurs for Acowldt W would li to rss our tas to t rfr for dtaild sustios ad corrctios o t auscrit. rsarc is suortd b t Natioal Natural Scic Foudatio of Cia (Grat No ). Rfrcs [] R. P. aba Four squars ad a -t owr Quart.. at. Oford (9) 9-0. [].. irc ad H. Davort O a tor of Davort ad Hilbro Acta atatica 00 (98) [] H. Davort ad H. Hilbro O idfiit quadratic fors i fiv variabls oural of Lodo at. Soc. (96) 8-9. [] A. Gos distributio -69. α odulo o Proc. Lodo at. Soc. (98)

16 6 [] G. Hara valus of trar quadratic fors at ri aruts atatia (00) [6] Wai Lau ad i-cit Liu Aroiatio b four squars ad a -t owr Soutast Asia ull. at. (978) -6. [7] W. P. Li ad. Z. Wa Dioati aroiatio wit o ri ad tr squars of ris Raaua oural (i rss). [8] R. C. Vaua Dioati aroiatio b ri ubrs Proc. Lodo at. Soc. 8 (97) 7-8. [9] R. C. Vaua Dioati aroiatio b ri ubrs Proc. Lodo at. Soc. 8 (97) 8-0.

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Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform. Discrete Fourier Transform Discrt Fourir Trasform Dfiitio - T simplst rlatio btw a lt- squc x dfid for ω ad its DTFT X ( ) is ω obtaid by uiformly sampli X ( ) o t ω-axis btw ω < at ω From t dfiitio of t DTFT w tus av X X( ω ) ω