Waring and Waring-Goldbach subbases with prescribed representation function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909012839104512 |
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| author | Táfula, Christian |
| author_facet | Táfula, Christian |
| contents | Let $h\geq 2$. For $A\subseteq \mathbb{N}$ write
\[ r_{A,h}(n) := \#\{(x_1,\ldots,x_h)\in A^h ~|~ x_1+\cdots+x_h=n\}. \]
We prove a general probabilistic subbasis principle: assuming an asymptotic for a weighted $h$-fold representation sum over a basis $B$, there exist subbases $A\subseteq B$ whose representation function $r_{A,h}(n)$ has prescribed regularly varying growth.
We apply this to $k$-th powers $\mathbb{N}^k$ and to $k$-th powers of primes $\mathbb{P}^k$. For $h \geq k^2-k+O(\sqrt{k})$, we show that every regularly varying function $F$ with $F(x)/\log x\to\infty$ in the admissible range is realized, with the expected singular series factor. In particular, there exists $A\subseteq \mathbb{N}^k$ such that
\[ r_{A,h}(n)\sim \mathfrak{S}_{k,h}(n) F(n). \]
Moreover, in the prime setting we obtain thin subbases $A\subseteq \mathbb{P}^k$ with $r_{A,h}(n)\asymp \log n$ for $n$ in the admissible congruence classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08371 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Waring and Waring-Goldbach subbases with prescribed representation function Táfula, Christian Number Theory Combinatorics 11P05, 11P32, 11B13, 05D40, 26A12 Let $h\geq 2$. For $A\subseteq \mathbb{N}$ write \[ r_{A,h}(n) := \#\{(x_1,\ldots,x_h)\in A^h ~|~ x_1+\cdots+x_h=n\}. \] We prove a general probabilistic subbasis principle: assuming an asymptotic for a weighted $h$-fold representation sum over a basis $B$, there exist subbases $A\subseteq B$ whose representation function $r_{A,h}(n)$ has prescribed regularly varying growth. We apply this to $k$-th powers $\mathbb{N}^k$ and to $k$-th powers of primes $\mathbb{P}^k$. For $h \geq k^2-k+O(\sqrt{k})$, we show that every regularly varying function $F$ with $F(x)/\log x\to\infty$ in the admissible range is realized, with the expected singular series factor. In particular, there exists $A\subseteq \mathbb{N}^k$ such that \[ r_{A,h}(n)\sim \mathfrak{S}_{k,h}(n) F(n). \] Moreover, in the prime setting we obtain thin subbases $A\subseteq \mathbb{P}^k$ with $r_{A,h}(n)\asymp \log n$ for $n$ in the admissible congruence classes. |
| title | Waring and Waring-Goldbach subbases with prescribed representation function |
| topic | Number Theory Combinatorics 11P05, 11P32, 11B13, 05D40, 26A12 |
| url | https://arxiv.org/abs/2501.08371 |