关于m_h（A_k）的一个注记

摘要：

A_k = {1，a_2，.，a_k}是n的h-基，如果每个不超过n的正整数都可以表示为不超过h个值a_i的和，记为n = n_h（A_k）.一个极值h-基A_k是一个使n尽可能大的基，则我们记为n = n_h（k）。“局部”邮票问题涉及特定集合A_k的性质，显然，n_h（A_k）不超过a_k的集合是不重要的。我们定义h_0（k）为n_h（A_k）超过a_k的h的最小值，这样的集合称为“容许”集合。我们说一个值n可以由A_k“生成”，如果它可以被表示为不超过h个值a_i的和，或者-等价地-如果它可以被表示为来自集合A '_k = {0，a_1，a_2，. a_k}。不能生成大于ha_k的值，我们现在考虑没有生成的小于小于ha_k的值的数量，记为m_h（A_k）-本质上是“间隙”数量的计数（参见Challis [1]和Selmer [5]第3.1页）。很容易证明，对于某个超过h_0（k）的值h_2（k），差m_h（A_k）- m_（h+1）（A_k）保持不变--也就是说，ha_k和（h+1）a_k之间缺失值的“模式”不随h的增加而改变。这里我们感兴趣的是h_0和h_2之间的缺失值的模式。在Selmer [5]的7.8页上，他证明了当h从h_0到h_2时，差序列m_h（A_k）- m_（h+1）（A_k）是非递增的。当我遇到这个猜想我不能说服自己，这可能是真的，发现一个可能的错误塞尔默的理由。我于1995年11月写信给他，1996年初他回复了我，同意情况可能是这样，并希望我能够找到一个相反的例子。本文记录了我成功寻找的一个反例，最终在1999年底找到。

英文摘要：

A_k = {1, a_2, ..., a_k} is an h-basis for n if every positive integer not exceeding n can be expressed as the sum of no more than h values a_i; we write n = n_h(A_k). An extremal h-basis A_k is one for which n is as large as possible, and then we write n = n_h(k). The "local" Postage Stamp Problem is concerned with properties of particular sets A_k, and it is clear that sets where n_h(A_k) does not exceed a_k are of little interest. We define h_0(k) to be the smallest value of h for which n_h(A_k) exceeds a_k; such sets are called "admissible". We say that a value n can be "generated" by A_k if it can be expressed as the sum of no more than h values a_i, or - equivalently - if it can be expressed as the sum of exactly h values a_i from the set A'_k = {0, a_1, a_2, ... a_k}. No values greater than ha_k can be generated, and we now consider the number of values less than ha_k that have no generation, denoted m_h(A_k) - essentially a count of the number of "gaps" (see Challis [1], and Selmer [5] page 3.1). It is easy to show that for some value h_2(k) exceeding h_0(k) the difference m_h(A_k) - m_(h+1)(A_k) remains constant - that is, the "pattern" of missing values between ha_k and (h+1)a_k does not change as h increases. Here we are interested in the pattern of missing values for values that lie between h_0 and h_2. On page 7.8 of Selmer [5] he conjectures that the sequence of differences m_h(A_k) - m_(h+1)(A_k) is non-increasing as h runs from h_0 to h_2. When I came across this conjecture I could not convince myself that it was likely to be true, having found a possible error in Selmer's justification. I wrote to him in November 1995, and early in 1996 he replied, agreeing that this might be the case and hoping that I might be able to find a counter example. This paper records my successful search for a counter example, eventually found late in 1999.

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