Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sums of linear transformations   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 many
choices of $\mathcal{L}_1$ and $\mathcal{L}_2$. As an application, we prove a
lower bound for $|A + \lambda \cdot A|$ when $A$ is a finite set of real
numbers and $\lambda$ is an algebraic number. In particular, when $\lambda$ 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 $$|A + \lambda \cdot
A| \geq (p^{1/d} + q^{1/d})^d |A| - o(|A|).$$ This is again best possible up to
the lower-order term and extends a recent result of Krachun and Petrov which
treated the case $\lambda = \sqrt{2}$.

12
The categories of this paper are: nucl-th
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
14
The categories of this paper are: math-ph