有理方体与堆垒数论中若干问题
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摘要
本文研究有理方体与堆垒数论中的一些问题。得到的主要结果如下：
1．棱长和面对角线长均为有理数的长方体称为有理方体。找有理方体等价于解丢番图方程组：x~2+y~2=l~2，x~2+z~2=m~2，y~2+z~2=n~2。许多数学家(包括欧拉)都曾研究过这一问题，给出了许多参数解。对任一本原商高组(a，b，c)，是否存在整数s，t使得s~2+t~2a~2，s~2+t~2b~2均为平方数呢?本文提出这一问题，并给出处理这个问题一般方法。一定意义上说，我们将这一问题归结为有限计算。
2．设A是k个正整数构成的集合。称{∑_(b∈B)b：B(?)A，B≠φ}为A的子集和集合，记为S(A)．M．B．Nathanson在Trans．Am．Math．Soc．(1995，347(4)：1409-1418)中研究了子集和的逆问题，并提出一个未解决问题。本文研究这一问题，提出如下猜想：若|A|=k≥ 6，gcd(A)=1，A中最大数为M，则同时给出这个猜想的部分证明。所得定理改进了Nathanson的结果。
3．设G为有限阿贝耳群。对满足条件|A|+|B|=|G|的G的非空子集A和B，我们确定了和集A+B={a+b：a∈A，b∈B}基数的所有可能取值，并描述使A+B≠G的那些子集A，B的结构。记
A(?)B={a+b：a∈A，b∈B，a≠b}，L(G)=|{g：g∈G，2g=0}|。对满足|A|+|B|=|G|+L(G)的G的非空子集A和B，我们证明了|A(?)B|≥|G|-2，并完全描述使A(?)B≠G的那些子集A，B的结构。所得结果推广了L．Gallardo等人在Z／nZ中关于A(?)A的相应结论(J．London Math．Soc．2002，65(2)：513-523)。
4．设A，B为整数区间[1，n]的两个非空子集，t为正整数。定义
(A+B)_t={x∈A+B：至少存在t对(a，6)∈A×B使得x=a+b}。本文证明了：若|A|+|B|≥(4n+4t-3)／3，则(A+B)_t中包含一个长至少为|A|+|B|-2t+1连续整数块。利用我们的方法，我们还证明了：若|A|+|B|≥4n／3，则A+B中包含长为n的算术级数；对任一满足2≤r≤(4n-1)／3的整数r，存在[1，n]的两个非空子集A，B使得|A|+|B|=r，而A+B中算术级数长度至多为(r+2)／2，进而至多为(2n-1)／3+1。同时，我们给出了这些结果在A(?)B中的平行结论。
In this dissertation, we investigate rational cuboids and some problems in additive number theory. The main results are summarized as follows.
1. A rational cuboid is a rectangular parallelepiped whose edges and face diagonals all have rational lengths. The problem of finding rational cuboids has attracted much historical interest. Are there rational cuboids with a given face? We pose this problem and develop a general theory to deal with this problem. In a sense, we reduce this problem to a finite calculation.
1．棱长和面对角线长均为有理数的长方体称为有理方体。找有理方体等价于解丢番图方程组：x~2+y~2=l~2，x~2+z~2=m~2，y~2+z~2=n~2。许多数学家(包括欧拉)都曾研究过这一问题，给出了许多参数解。对任一本原商高组(a，b，c)，是否存在整数s，t使得s~2+t~2a~2，s~2+t~2b~2均为平方数呢?本文提出这一问题，并给出处理这个问题一般方法。一定意义上说，我们将这一问题归结为有限计算。
2．设A是k个正整数构成的集合。称{∑_(b∈B)b：B(?)A，B≠φ}为A的子集和集合，记为S(A)．M．B．Nathanson在Trans．Am．Math．Soc．(1995，347(4)：1409-1418)中研究了子集和的逆问题，并提出一个未解决问题。本文研究这一问题，提出如下猜想：若|A|=k≥ 6，gcd(A)=1，A中最大数为M，则同时给出这个猜想的部分证明。所得定理改进了Nathanson的结果。
3．设G为有限阿贝耳群。对满足条件|A|+|B|=|G|的G的非空子集A和B，我们确定了和集A+B={a+b：a∈A，b∈B}基数的所有可能取值，并描述使A+B≠G的那些子集A，B的结构。记
A(?)B={a+b：a∈A，b∈B，a≠b}，L(G)=|{g：g∈G，2g=0}|。对满足|A|+|B|=|G|+L(G)的G的非空子集A和B，我们证明了|A(?)B|≥|G|-2，并完全描述使A(?)B≠G的那些子集A，B的结构。所得结果推广了L．Gallardo等人在Z／nZ中关于A(?)A的相应结论(J．London Math．Soc．2002，65(2)：513-523)。
4．设A，B为整数区间[1，n]的两个非空子集，t为正整数。定义
(A+B)_t={x∈A+B：至少存在t对(a，6)∈A×B使得x=a+b}。本文证明了：若|A|+|B|≥(4n+4t-3)／3，则(A+B)_t中包含一个长至少为|A|+|B|-2t+1连续整数块。利用我们的方法，我们还证明了：若|A|+|B|≥4n／3，则A+B中包含长为n的算术级数；对任一满足2≤r≤(4n-1)／3的整数r，存在[1，n]的两个非空子集A，B使得|A|+|B|=r，而A+B中算术级数长度至多为(r+2)／2，进而至多为(2n-1)／3+1。同时，我们给出了这些结果在A(?)B中的平行结论。
In this dissertation, we investigate rational cuboids and some problems in additive number theory. The main results are summarized as follows.
1. A rational cuboid is a rectangular parallelepiped whose edges and face diagonals all have rational lengths. The problem of finding rational cuboids has attracted much historical interest. Are there rational cuboids with a given face? We pose this problem and develop a general theory to deal with this problem. In a sense, we reduce this problem to a finite calculation.
引文
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[17] S. Eliahou, M. Kervaire. Restricted sums of sets of cardinality 1 + p in a vector space over F_p. Discrete Math., 235(2001), 199-213.
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[2] M. B. Nathanson. Elementary methods in number theory. Grad. Texts in Math. 195, Springer, 2000.
[3] 华罗庚．数论导引．北京，科学出版社，1957．
[4] 华罗庚．堆垒素数论．北京：科学出版社．1953．
[5] 陈景润著．陈景润文集．南昌：江西教育出版社，1998．
[6] 李文林(主编)．王元论哥德巴赫猜想．济南：山东教育出版社．1999．
[7] 王元．潘承彪(主编)．潘承洞文集．济南：山东教育出版社，2002．
[8] 潘承洞，潘承彪．解析致论基础．北京，科学出版社，1991．
[9] R．K．盖伊著，张明尧译．数论中未解决的同题．北京：科学出版社，2003．
[10] M. B. Nathanson. Additive number theory: inverse problems and the geometry of sumsets. Grad. Texts in Math. 165, Springer, 1996.
[11] G. A. Freiman. Structure theory of set addition. Astérisque, 258(1999),1-33.
[12] G. A. Freiman. Foundations of a structural theory of set addition (Kazan. 1966)(Russian); Translations of mathematical Monographs 37 (American Mathematical Society, Providence, RI,1973)(English).
[13] I. Z. Ruzsa. Arithmetical progressions and the number of sums. Periodica Math. Hung. 25(1992), 105-111.
[14] G. Elekes, M. B. Nathanson, I. Z. Ruzsa. Convexity and sumsets. J. Number Theory, 83(2)(2000), 194-201.
[15] A. S. Fainleib. On sumsets in Euclidean space. J. Number Theory, 82(2000), 121-133.
[16] S. Eliahou, M. Kervaire. Sumsets in vector spaces over finite fields. J. Number Theory, 71(1)(1998), 12-39.
[17] S. Eliahou, M. Kervaire. Restricted sums of sets of cardinality 1 + p in a vector space over F_p. Discrete Math., 235(2001), 199-213.
[18] I. Z. Ruzsa. Sumsets of Sidon sets. Acta Arith., 77(4)(1996), 353-359.
[19] Y. G. Chen. On addition of two sets of integers. Acta Arith., 80(1997), 83-87.
[20] T. Schoen. On finite difference sets. Acta Math. Hungar., 79(1-2)(1998), 123-138
[21] Y. Stanchescu. Sumsets in vector spaces over finite fields. J. Number Theory, 71(1)(1998), 12-39.
[22] Y. V. Stanchescu. An upper bound for d-dimensional difference sets. Combinatorica, 21 (4) (2001),591-595.
[23] Y. V. Stanchescu. Planar sets containing no three collinear points and nonaveraging sets of integers. Discrete Math., 256(2002),387-395.
[24] V. F. Lev. Structure theorem for multiple addition and the Frobenius problem. J. Number Theory, 58(1996), 79-88.
[25] V. F. Lev. Optimal representations by sumsets and subset sums. J. Number theory, 62(1)(1997), 127-143.
[26] P. Erds, M. B. Nathanson, A. Sárkzy. Sumsets containing infinite arithmetic progressions. J. Number Theory, 28(1988), 159-166.
[27] B. Green. Arithmetic progressions in sumsets. Geom. Funct. Anal., 12(3)(2002), 584-597.
[28] V. F. Lev. Addendum to "structure theorem for multiple addition". J. Number Theory, 65(1997), 96-100.
[29] V. F. Lev. Blocks and progressions in subset sum sets. Acta Arith. 106(2)(2003), 123-142.
[30] I. Z. Ruzsa. Arithmetic progressions in sumsets. Acta Arith., 60(2)(1991), 191-202.
[31] Y. G. Chen, B. Wang. On additive pwperties of two special sequence. Acta Arith., 110(3)(2003), 299-303.
[32] Y. G. Chen. On sums and products of integers. Proc. Amer. Math. Soc., 127(1999),1927-1933.
[33] Y. G. Chen. On sums and intersects of sequences. Discrete Math., 223(2000), 351-354.
[34] G. Dombi. Additive properties of certain sets. Acta Arith., 103(2)(2002), 137-146.
[35] G. A. Freiman. Inverse problems in additive number theory. U. Zap. Kazan Univ., 115(14)(1955),109-115[Russian]
[36] G. A. Freiman. Inverse problems in additive theory of numbers . Izv. Acad. Nauk SSSR Ser. Mat., 19(1955),275-284[Russian].
[37] Y. O. Hamidoune, A. Plagne. A generalization of Freiman's 3κ - 3 theorem. Acta Arith., 103(2)(1999), 147-156.
[38] Y. O. Hamidoune, φ. J. Rφdseth. A inverse theorem rood p. Acta Arith., XCⅡ(3)(1999), 251-262.
[39] Y. O. Hamidoune. Some results in additive number theory Ⅰ: the critical pair theory. Acta Arith., XCⅣ(2)(2000), 97-119.
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