论文标题
线性转换的总和
Sums of linear transformations
论文作者
论文摘要
我们表明，如果$ \ MATHCAL {L} _1 $和$ \ MATHCAL {L} _2 $是从$ \ Mathbb {Z}^d $到$ \ Mathbb {z}^d $的线性转换，那么，满足某些有限的条件，那么，对于任何有限的$ a $ a $ a $ a $ a $ a of of of $ a a $ a a $ a a $ \ a z} z} $ | \ MATHCAL {l} _1 A+\ MATHCAL {L} _2 A | \ GEQ \ left（| \ det（\ Mathcal {l} _1）|^{1/D}+| \ det（\ MATHCAL {\ MATHCAL {l} _2 _2）结果纠正并确认了bukh的猜想的两夏和情况，最好直至低阶术语，以获得$ \ Mathcal {l} _1 $和$ \ Mathcal {l} _2 _2 $的某些选择。作为一个应用程序，当$ a $是有限的实际数字，而$λ$是代数号码时，我们证明了$ | a +λ\ cdot a | $的下限。特别是，当$λ$的形式为$（p/q）^{1/d} $，对于某些$ p，q，q，in \ mathbb {n} $，每个人都尽可能小，我们表明$$ | a +λ\ cdot a | \ geq（p^{1/d} + q^{1/d}）^d | a | -o（| a |）。$$这再次可能是较低阶段，并扩展了Krachun和Petrov的最新结果，该结果治疗了$λ= \ sqrt {2} $。
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 $λ= \sqrt{2}$.
