论文标题
代数$λ$的$ a+λa$的大小
On the size of $A+λA$ for algebraic $λ$
论文作者
论文摘要
对于有限的集合$ a \ subset \ mathbb {r} $和真实的$λ$，让$ a+λa：= \ {a+λb：\ \，a，a，b \ in a \} $。将Freiman的结构定理与少量的常数与Prékopa的离散类似物结合在一起 - Leindler不等式，我们证明了下限的$ | a+| a+\ sqrt {2} a | \ geq（1+ \ sqrt {1+\ sqrt {2}} {2}）本质上很紧。我们还对任意代数$λ$的$ \ liminf | a+λa|/| a | $的值进行了猜想。最后，对于给定的线性操作员$ k+\ \ \ \ \ \ \ \ \ mathcal $ \ mathcal {t} \ in \ operatorNAMe {end}（\ mathbb {r}^d）$和一个compact set $ k \ subset $ kathbb {r r} d $，我们证明了$ k+\ mathcal {t} k $的$ k+\ \ \ \ \ mathcal {t} k $的紧密下限。这种连续的结果支持猜想，并产生其中的上限。
For a finite set $A\subset \mathbb{R}$ and real $λ$, let $A+λA:=\{a+λb :\, a,b\in A\}$. Combining a structural theorem of Freiman on sets with small doubling constants together with a discrete analogue of Prékopa--Leindler inequality we prove a lower bound $|A+\sqrt{2} A|\geq (1+\sqrt{2})^2|A|-O({|A|}^{1-\varepsilon})$ which is essentially tight. We also formulate a conjecture about the value of $\liminf |A+λA|/|A|$ for an arbitrary algebraic $λ$. Finally, we prove a tight lower bound on the Lebesgue measure of $K+\mathcal{T} K$ for a given linear operator $\mathcal{T}\in \operatorname{End}(\mathbb{R}^d)$ and a compact set $K\subset \mathbb{R}^d$ with fixed measure. This continuous result supports the conjecture and yields an upper bound in it.
