Sums of linear transformations
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913581292847104 |
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| author | Conlon, David Lim, Jeck |
| author_facet | Conlon, David Lim, Jeck |
| contents | We show that if $\mathcal{L}_1$ and $\mathcal{L}_2$ are linear transformations from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ satisfying certain mild conditions, then, for any finite subset $A$ of $\mathbb{Z}^d$, $$|\mathcal{L}_1 A+\mathcal{L}_2 A|\geq \left(|\det(\mathcal{L}_1)|^{1/d}+|\det(\mathcal{L}_2)|^{1/d}\right)^d|A|- o(|A|).$$ This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of $\mathcal{L}_1$ and $\mathcal{L}_2$. As an application, we prove a lower bound for $|A + λ\cdot A|$ when $A$ is a finite set of real numbers and $λ$ is an algebraic number. In particular, when $λ$ is of the form $(p/q)^{1/d}$ for some $p, q, d \in \mathbb{N}$, each taken as small as possible for such a representation, we show that $$|A + λ\cdot A| \geq (p^{1/d} + q^{1/d})^d |A| - o(|A|).$$ This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $λ= \sqrt{2}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_09827 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sums of linear transformations Conlon, David Lim, Jeck Combinatorics Number Theory 05D99, 11B13, 11B75, 11B30 We show that if $\mathcal{L}_1$ and $\mathcal{L}_2$ are linear transformations from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ satisfying certain mild conditions, then, for any finite subset $A$ of $\mathbb{Z}^d$, $$|\mathcal{L}_1 A+\mathcal{L}_2 A|\geq \left(|\det(\mathcal{L}_1)|^{1/d}+|\det(\mathcal{L}_2)|^{1/d}\right)^d|A|- o(|A|).$$ This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of $\mathcal{L}_1$ and $\mathcal{L}_2$. As an application, we prove a lower bound for $|A + λ\cdot A|$ when $A$ is a finite set of real numbers and $λ$ is an algebraic number. In particular, when $λ$ is of the form $(p/q)^{1/d}$ for some $p, q, d \in \mathbb{N}$, each taken as small as possible for such a representation, we show that $$|A + λ\cdot A| \geq (p^{1/d} + q^{1/d})^d |A| - o(|A|).$$ This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $λ= \sqrt{2}$. |
| title | Sums of linear transformations |
| topic | Combinatorics Number Theory 05D99, 11B13, 11B75, 11B30 |
| url | https://arxiv.org/abs/2203.09827 |