Representation functions of sequences in additive number theory
Representation functions of sequences in additive number theory
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DOI：
10.1090/s0002-9939-1978-0503522-6
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发表时间：
1978
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影响因子：
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通讯作者：
M. Nathanson
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文献类型：
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作者：
M. Nathanson
让吗?是一个非负整数集,让r2f (n)表示的数量表示的n n =与人工智能ai + aj, aj E d。一组6周期如果^ E E意味着+ m E El m >我和所有> n它证明如果E不是周期性的,然后每组' + # E存在无穷多n, r2 (n) # r2 (n)。此外，构造了除有限个n外的所有周期集El和S对满足r2(n) = r2s(n)。设6l为非负整数集。设r^(n)表示n作为6d中h个元素的和的表示形式的个数。如果f(z) = E-a EqZa是6'的生成函数，则f(z)h = 0？=Orha(n) zn。设rd(n)表示n作为任意数目的61元素的和的表示的个数。如果0 M C，则rg(n) = XO Ir d(n)对所有n都是有限的。许多作者已经研究了表示函数[147]。在这篇文章中，我考虑的问题是：序列r (n)和rd(n)在多大程度上决定了这个集合？我将证明如果D和'9i3是非负整数的集合使得r^ (n) = r^ (n)对于某些h >和所有n >，或者如果ro(n) = rB(n)对于所有n >，那么6，= fi3。然而，确实存在6组！和'iJ使得9 (n) = r4(n)对于所有足够大的n，但Cl7:#3。所有这样的集合6C和93对将被显式构造。一个无限整数集6C是周期的，如果存在整数m > 1和N，使得对于所有一个> N， a E cT意味着a + m E 6S。将证明，如果集合C不是周期的，那么对于每一个集合0J =# cT，对于无限多个N，我们必须有r(N) 7# r4(N)。定理1。设d（和93）是一组非负整数，设rh(n)和rhs(n)分别表示n作为C和(M)中h个元素的和的表示个数。如果rh(n) = rhB(n)对于所有n > 0，则(E =(3)。证明。如果rhd(n) = rh4(n)对于所有n > 0，则
Let ? be a set of nonnegative integers, and let r2f(n) denote the number of representations of n in the form n = ai + aj with ai, aj E d. The set 6 is periodic if a^E E implies a + m E El for some m > I and all a > N. It is proved that if E is not periodic, then for every set ' +# E there exist infinitely many n such that r2(n) # r2s(n). Moreover, all pairs of periodic sets El and S are constructed that satisfy r2(n) = r2s(n) for all but finitely many n. Let 6l be a set of nonnegative integers. Let r^(n) denote the number of representations of n as a sum of h elements of 6d. If f(z) = E-a EqZa is the generating function for 6', then f(z)h = 0?=Orha(n)z n. Let rd(n) denote the number of representations of n as a sum of an arbitrary number of elements of 61. If 0 M C, then rg(n) = XO Ir d(n) is finite for all n. Representation functions have been studied by various authors [147]. In this note I consider the question: To what extent do the sequences r, (n) and rd(n) determine the set ? I shall prove that if D and '9i3 are sets of nonnegative integers such that r^ (n) = r^ (n) for some h > 1 and all n > 0, or if ro(n) = rB(n) for all n > 0, then 6, = fi3. However, there do exist sets 6! and 'iJ such that 9,(n) = r4(n) for all sufficiently large n, but Cl7 :#3 . All such pairs of sets 6C and 93 will be constructed explicitly. An infinite set 6C of integers is called periodic if there exist integers m > 1 and N such that a E cT implies a + m E 6S for all a > N. It will be shown that if the set C is not periodic, then for every set 0J =# CT we must have r(n) 7# r4(n) for infinitely many n. THEOREM 1. Let d( and 93 be sets of nonnegative integers, and let rh(n) and rhs(n) denote the number of representations of n as a sum of h elements of C and (M, respectively. If rh(n) = rhB(n) for all n > 0, then (E = (3. PROOF. If rhd(n) = rh4(n) for all n > 0, then