For $A \subset \mathbb{N}$ and positive integer $h > 0$, define $r_{A,h}(n)$ to be the number of ways to write $n$ as the sum of $h$ (not necessarily distinct) elements of $A$. We say $A$ is an additive basis of order $h$ if $r_{A,h}(n) > 0$ for all $n$ sufficiently large. Well-known additive bases of finite order include the Waring bases ($\mathbb{N}^k$ for some $k > 0$) and the primes (Goldbach-Shrinel'man Theorem, the $h = 4$ case known as Vinogradov's Theorem). In both cases, the COUNTING function for $A$, which is the function $f(n) = |A \cap [1,n]|$, is quite regular. In particular, for the Waring bases we have $f(n) \sim n^{1/k}$ and for the primes we have $f(n) = \pi(n) \sim \frac{n}{\log(n)}$ (Prime Number Theorem). My question is are there any well-known cases where $f(n)$ is very irregular? In other words there exists some function $h(n)$ g(n)$ and constants $0 < c_1 < c_2 < 1$ such that $\displaystyle \liminf_{n \rightarrow \infty} f(n)/g(n) < c_1, \limsup_{n \rightarrow \infty} f(n)/g(n) > c_2$. I am interested in naturally occurring examples, and less interested in a specific construction of a basis with this property (that shouldn't be too difficult, though I have not tried yet). In particular, whether this property is worthy of serious study.
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Known additive bases with irregular counting functionFor $A \subset \mathbb{N}$ and positive integer $h > 0$, define $r_{A,h}(n)$ to be the number of ways to write $n$ as the sum of $h$ (not necessarily distinct) elements of $A$. We say $A$ is an additive basis of order $h$ if $r_{A,h}(n) > 0$ for all $n$ sufficiently large. Well-known additive bases of finite order include the Waring bases ($\mathbb{N}^k$ for some $k > 0$) and the primes (Goldbach-Shrinel'man Theorem, the $h = 4$ case known as Vinogradov's Theorem). In both cases, the COUNTING function for $A$, which is the function $f(n) = |A \cap [1,n]|$, is quite regular. In particular, for the Waring bases we have $f(n) \sim n^{1/k}$ and for the primes we have $f(n) = \pi(n) \sim \frac{n}{\log(n)}$ (Prime Number Theorem). My question is are there any well-known cases where $f(n)$ is very irregular? In other words there exists some function $h(n)$ and constants $0 < c_1 < c_2 < 1$ such that $\displaystyle \liminf_{n \rightarrow \infty} f(n)/g(n) < c_1, \limsup_{n \rightarrow \infty} f(n)/g(n) > c_2$. I am interested in naturally occurring examples, and less interested in a specific construction of a basis with this property (that shouldn't be too difficult, though I have not tried yet). In particular, whether this property is worthy of serious study.
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