arXiv subjects

Albert Bush

Publications and source records attributed to Albert Bush.

4 recordsLinked to original sources

arXiv2016

New Upper Bound for Sums of Dilates

For $\lambda \in \mathbb{Z}$, let $\lambda \cdot A = \{ \lambda a : a \in A\}$. Suppose $r, h\in \mathbb{Z}$ are sufficiently large and comparable to each other. We prove that if $|A+A| \le K |A|$ and $\lambda_1, \ldots, \lambda_h \le 2^r$, then \[ |\lambda_1 \cdot A + \ldots + \lambda_h \cdot A | \le K^{ 7 rh /\ln (r+h) } |A|. \] This improves upon a result of Bukh who shows that \[ |\lambda_1 \cdot A + \ldots + \lambda_h \cdot A | \le K^{O(rh)} |A|. \] Our main technique is to combine Bukh's idea of considering the binary expansion of $\lambda_i$ with a result on biclique decompositions of bipartite graphs.lique decompositions.

math.CO↗

arXiv2014

Few products, many h-fold sums

Improving upon a technique of Croot and Hart, we show that for every $h$, there exists an $ε> 0$ such that if $A \subseteq \mathbb{R}$ is sufficiently large and $|A.A| \le |A|^{1+ε}$, then $|hA| \ge |A|^{Ω(e^{\sqrt{c\log{h}}})}$.

math.CO↗

arXiv2014

Minimum degree thresholds for bipartite graph tiling

For any bipartite graph $H$, we determine a minimum degree threshold for a balanced bipartite graph $G$ to contain a perfect $H$-tiling. We show that this threshold is best possible up to a constant depending only on $H$. Additionally, we prove a corresponding minimum degree threshold to guarantee that $G$ has an $H$-tiling missing only a constant number of vertices. Our threshold for the perfect tiling depends on either the chromatic number $χ(H)$ or the critical chromatic number $χ_{cr}(H)$ while the threshold for the almost perfect tiling only depends on $χ_{cr}(H)$. Our results answer two questions of Zhao. They can be viewed as bipartite analogs to the results of Kuhn and Osthus and of Shokoufandeh and Zhao.

math.CO↗

arXiv2014

Order-preserving Freiman isomorphisms

An order-preserving Freiman 2-isomorphism is a map $\phi:X \rightarrow \mathbb{R}$ such that $\phi(a) < \phi(b)$ if and only if $a < b$ and $\phi(a)+\phi(b) = \phi(c)+\phi(d)$ if and only if $a+b=c+d$ for any $a,b,c,d \in X$. We show that for any $A \subseteq \mathbb{Z}$, if $|A+A| \le K|A|$, then there exists a subset $A' \subseteq A$ such that the following holds: $|A'| \gg_K |A|$ and there exists an order-preserving Freiman 2-isomorphism $\phi: A' \rightarrow [-c|A|,c|A|] \cap \mathbb{Z}$ where $c$ depends only on $K$. Several applications are also presented.

math.CO↗