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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.03690 |
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| _version_ | 1866929746945769472 |
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| author | Mansfield, Samuel |
| author_facet | Mansfield, Samuel |
| contents | Let $\mathcal{F}$ be a set of $n$ real analytic functions with linearly independent derivatives restricted to a compact interval $I$. We show that for any finite set $A \subset I$, there is a function $f \in \mathcal{F}$ that satisfies
$$|2^{n-1}f(A)-(2^{n-1}-1)f(A)|\gg_{\mathcal{F},I} |A|^{ϕ(n)},$$ where $ϕ:\mathbb{N} \to \mathbb{R}$ satisfies the recursive formula
$$ϕ(1)=1, \quad ϕ(n)=1+\frac{1}{1+\frac{1}{ϕ(n-1)}} \quad \text{for } n\geq 2.$$ The above result allows us to prove the bound
$$|2^nf(A-A)-(2^n-1)f(A-A)| \gg_{f,n,I} |A|^{1+ϕ(n)}$$ where $f$ is an analytic function for which any $n$ distinct non-trivial discrete derivatives of $f'$ are linearly independent. This condition is satisfied, for instance, by any polynomial function of degree $m \geq n+1$. We also check this condition for the function $\arctan(e^x)$ with $n=3$, allowing us to improve upon a recent bound on the additive growth of the set of angles in a Cartesian product due to Roche-Newton. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03690 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Additive growth amongst images of linearly independent analytic functions Mansfield, Samuel Combinatorics Number Theory 11B13, 11B30 Let $\mathcal{F}$ be a set of $n$ real analytic functions with linearly independent derivatives restricted to a compact interval $I$. We show that for any finite set $A \subset I$, there is a function $f \in \mathcal{F}$ that satisfies $$|2^{n-1}f(A)-(2^{n-1}-1)f(A)|\gg_{\mathcal{F},I} |A|^{ϕ(n)},$$ where $ϕ:\mathbb{N} \to \mathbb{R}$ satisfies the recursive formula $$ϕ(1)=1, \quad ϕ(n)=1+\frac{1}{1+\frac{1}{ϕ(n-1)}} \quad \text{for } n\geq 2.$$ The above result allows us to prove the bound $$|2^nf(A-A)-(2^n-1)f(A-A)| \gg_{f,n,I} |A|^{1+ϕ(n)}$$ where $f$ is an analytic function for which any $n$ distinct non-trivial discrete derivatives of $f'$ are linearly independent. This condition is satisfied, for instance, by any polynomial function of degree $m \geq n+1$. We also check this condition for the function $\arctan(e^x)$ with $n=3$, allowing us to improve upon a recent bound on the additive growth of the set of angles in a Cartesian product due to Roche-Newton. |
| title | Additive growth amongst images of linearly independent analytic functions |
| topic | Combinatorics Number Theory 11B13, 11B30 |
| url | https://arxiv.org/abs/2503.03690 |