リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

大学・研究所にある論文を検索できる 「Arithmetic progressions of Piatetski-Shapiro sequences and related problems」の論文概要。リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。
リケラボ

コピーが完了しました

URLをコピーしました

論文の公開元へ論文の公開元へ
書き出し

Arithmetic progressions of Piatetski-Shapiro sequences and related problems

齋藤, 耕太 名古屋大学

2021.11.10

概要

This thesis is a summary of the works of the author. We discuss problems whether a given set contains arithmetic progressions or not. By Szemer´edi’s theorem, any subset of positive integers with positive upper asymptotic den- sity contains arbitrarily long arithmetic progressions. However, it is still difficult to find long arithmetic progressions of a sparse set. We mainly in- vestigate arithmetic progressions of Piatetski-Shapiro sequences and more general sequences which have upper asymptotic density 0. Here for every non-integral α > 1, the sequence of the integer parts of nα (n = 1, 2, . . .) is called the Piatetski-Shapiro sequence with exponent α, and let PS(α) be the set of those terms. We present three main results.

Firstly, in the case 1 < α < 2, we reveal the explicit density of the set of n N such that the integer parts of (n + rj)α (j = 0, 1, . . . , k 1) forms an arithmetic progression for all integers k 3 and r 1. This density is equal to 1/(k 1) which is independent of α and r. We get an extended result for more general sequences. This work is collaborated with Yuuya Yoshida.

Secondary, fix a, b, c N. For every 2 < s < t, we study the set of α [s, t] such that the Diophantine equation ax + by = cz has infinitely many solutions (x, y, z) PS(α)3 with x, y, z pairwise distinct. We show that the Hausdorff dimension of the set is greater than or equal to 1/s3. As a consequence, there are uncountably many α > 2 such that PS(α) contains infinitely many arithmetic progressions of length 3. This work is collaborated with Toshiki Matsusaka.

Thirdly, we investigate Diophantine linear equations with two variables in Piatetski-Shapiro sequences. Let a, b R with 0 b < a and a = 1 satisfying that y = ax + b has infinitely many solutions of positive integers. Then we reveal the Hausdorff dimension of the set of α [s, t] such that y = ax + b has infinitely many solutions (x, y) PS(α)2. This dimension is coincident with 2/s for all 2 < s < t. Furthermore, we show that for all 1 < α < 2, y = ax + b has infinitely many solutions (x, y) PS(α)2. As a consequence, we obtain a partial result of the existence of a perfect Euler brick.

論文の公開元へ

関連論文

関連論文をもっと見る

参考文献

[AG16] Y. Akbal and A. M. Gu¨loglu, Waring’s problem with Piatetski- Shapiro numbers, Mathematika, 62 (2016), 524–550.

[AZ84] G. I. Arkhipov and A. N. Zhitkov, Waring’s problem with nonin- tegral exponent, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), 1138– 1150.

[BKS19] V. Bergelson, G. Kolesnik, and Y. Son, Uniform distribution of sub- polynomial functions along primes and applications, J. Anal. Math. 137 (2019), 135–187.

[Bos81] M. Boshernitzan, An extension of Hardy’s class L of “orders of infinity”, J. Anal. Math. 39 (1981), 235–255.

[Bos82] M. Boshernitzan, New “orders of infinity”, J. Anal. Math. 41(1982), 130–167.

[Bos84a] M. Boshernitzan, Discrete “orders of infinity”, Amer. J. Math. 106(1984), 1147–1198.

[Bos84b] M. Boshernitzan, “Orders of infinity” generated by dierence equa- tions, Amer. J. Math. 106 (1984), 1067–1089.

[Bos87] M. Boshernitzan, Second order dierential equations over Hardy fields, J. London Math. Soc. (2), 35 (1987), 109–120.

[Bos94] M. Boshernitzan, Uniform distribution and Hardy fields, J. Anal.Math. 62 (1994), 225–240.

[Bou61] N. Bourbaki, Fonctions d’une variable r´eele, Chapitre V (E´tude Locale des Fonctions), second ed., Hermann, Paris, 1961.

[DM97] H. Darmon and L. Merel, Winding quotients and some variants of Fermat’s last theorem, J. Reine Angew. Math. 490 (1997), 81–100.

[D´en52] P. D´enes,U¨ber die Diophantische Gleichung xl + yl = czl, ActaMath. 88 (1952), 241–251.

[Des73] J. -M. Deshouillers, Probl`eme de Waring avec exposants non en- tiers, Bull. Soc. Math. France 101 (1973), 285–295.

[Des76] J. -M. Deshouillers, Nombres premiers de la forme [nc], C. R. Acad.Sci. Paris S´er. A-B, 282 (1976), Ai, A131–A133.

[Dic66] L. E. Dickson, History of the theory of numbers. Vol. II: Diophan- tine analysis, Chelsea Publishing Co., New York, 1966.

[DT97] M. Drmota and R. F. Tichy, Sequences, discrepancies and appli- cations, Lecture Notes in Mathematics, vol. 1651, Springer-Verlag, Berlin, 1997.

[ET36] P. Erd˝os and P. Tur´an, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.

[Fal14] K. Falconer, Fractal geometry, third ed., John Wiley & Sons, Ltd., Chichester, 2014, Mathematical foundations and applications.

[Fra09] N. Frantzikinakis, Equidistribution of sparse sequences on nilmani- folds, J. Anal. Math. 109 (2009), 353–395.

[FW09] N. Frantzikinakis and M. Wierdl, A Hardy field extension of Sze- mer´edi’s theorem, Adv. Math. 222 (2009), 1–43.

[Gla17] D. Glasscock, Solutions to certain linear equations in Piatetski- Shapiro sequences, Acta Arith. 177 (2017), 39–52.

[Gla20] D. Glasscock, A perturbed Khinchin-type theorem and solutions to linear equations in Piatetski-Shapiro sequences, Acta Arith. 192 (2020), 267–288.

[GRS90] R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey theory, second ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1990, A Wiley- Interscience Publication.

[GK91] S. W. Graham and G. Kolesnik, van der Corput’s method of expo- nential sums, London Mathematical Society Lecture Note Series, vol. 126, Cambridge University Press, Cambridge, 1991.

[Gre05] B. Green, Roth’s theorem in the primes, Ann. of Math. (2), 161(2005), 1609–1636.

[GT08] B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2), 167 (2008), 481–547.

[Gri96] S. A. Gritsenko, On estimates for trigonometric sums with respect to the third derivative, Mat. Zametki, 60 (1996), 383–389, 479.

[Har12] G. H. Hardy, Properties of logarithmico-exponential functions, Proc.London Math. Soc. (2), 10 (1912), 54–90.

[Har24] G. H. Hardy, Orders of infinity, second ed., Cambridge University Press, Cambridge, 1924.

[HB83]D. R. Heath-Brown, The Pjatecki˘ı-Sˇapiro prime number theorem,J. Number Theory, 16 (1983), 242–266.

[Kok50] J. F. Koksma, Some theorems on Diophantine inequalities, Scrip- tum no.5, Math. Centrum Amsterdam, 1950.

[Kol67]G. Kolesnik, The distribution of primes in sequences of the form

[nc], Mat. Zametki, 2 (1967), 117–128.

[Kol85] G. Kolesnik, Primes of the form [nc], Pacific J. Math. 118 (1985), 437–447.

[KN74] L. Kuipers and H. Niederreiter, Uniform distribution of se- quences, Wiley-Interscience [John Wiley & Sons], New York- London-Sydney,1974, Pure and Applied Mathematics.

[Lei80] D. Leitmann, Absch¨atzung trigonometrischer Summen, J. Reine Angew. Math. 317 (1980), 209–219.

[Lis02] Y. R. Listratenko, Estimation of the number of summands in War- ing’s problem for the integer parts of fixed powers of nonnegative numbers, vol. 3, 2002, Dedicated to the 85th birthday of Nikola˘ı Mikha˘ılovich Korobov (Russian), 71–77.

[LR92] H. Q. Liu and J. Rivat, On the Pjatecki˘ı-Sˇapiro prime number the- orem, Bull. London Math. Soc. 24 (1992), 143-147.

[MS20] T. Matsusaka and K. Saito, Linear Diophantine equations in Piatetski-Shapiro sequences, preprint, 2020, available at https:/ arxiv.org/abs/2009.12899 to appear in Acta Arithmetica.

[May15] J. Maynard, Small gaps between primes, Ann. of Math. (2), 181(2015), 383–413.

[Mir15] M. Mirek, Roth’s theorem in the Piatetski-Shapiro primes, Rev.Mat. Iberoam. 31 (2015), 617–656.

[PS53] I. I. Piatetski-Shapiro, On the distribution of prime numbers in sequences of the form [f (n)], Mat. Sbornik N.S. 33(75) (1953), 559–566.

[Pol14] D. H. J. Polymath, Variants of the Selberg sieve, and bounded in- tervals containing many primes, Res. Math. Sci. 1 (2014), Art. 12, 83.

[RS01] J. Rivat and P. Sargos, Nombres premiers de la forme nc , Canad.J. Math. 53 (2001), 414–433.

[Ros83a] M. Rosenlicht, Hardy fields, J. Math. Anal. Appl. 93 (1983), 297– 311.

[Ros83b] M. Rosenlicht, The rank of a Hardy field, Trans. Amer. Math. Soc.280 (1983), 659–671.

[Rot52]K. Roth, Sur quelques ensembles d’entiers, C. R. Acad. Sci. Paris,234 (1952), 388–390.

[Sai20] K. Saito, Linear equations with two variables in Piatetski- Shapiro sequences, preprint, 2020, available at https://arxiv.org/ abs/2011.05069, to appear in Acta Arith.

[SY19] K. Saito and Y. Yoshida, Arithmetic progressions in the graphs of slightly curved sequences, J. Integer Seq. 22 (2019), Art. 19.2.1, 25.

[SY21] K. Saito and Y. Yoshida, Distributions of finite sequences repre- sented by polynomials in Piatetski-Shapiro sequences, J. Number Theory, 222 (2021), 115–156.

[Sar95] P. Sargos, Points entiers au voisinage d’une courbe, sommes trigonom´etriques courtes et paires d’exposants, Proc. London Math. Soc. (3) 70 (1995), 285–312.

[Seg33] B. I. Segel, On a theorem analogous to Waring’s theorem, Dokl.Akad. Nauk SSSR, 1 (1933), 47–49.

[Sze75] E. Szemer´edi, On sets of integers containing no k elements in arith- metic progression, Acta Arith. 27 (1975), 199–245.

[Szu¨52] P. Szu¨sz, U¨ber ein Problem der Gleichverteilung, Akad´emiai Kiad´o, Budapest, 1952.

[Tit86] E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed., The Clarendon Press, Oxford University Press, New York, 1986, Edited and with a preface by D. R. Heath-Brown.

[Van27] B. L. Van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wiskd., II. Ser. 15 (1927), 212–216.

参考文献をもっと見る