Geodesic
New lower bounds for cardinalities of higher dimensional difference sets and sumsets
Discrete analysis (2022)
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Let $d \geq 4$ be a natural number and let $A$ be a finite, non-empty subset of $\mathbb{R}^d$ such that $A$ is not contained in a translate of a hyperplane. In this setting, we show that \[ |A-A| \geq \bigg(2d - 2 + \frac{1}{d-1} \bigg) |A| - O_{d}(|A|^{1- δ}), \] for some absolute constant $δ>0$ that only depends on $d$. This provides a sharp main term, consequently answering questions of Ruzsa and Stanchescu up to an $O_{d}(|A|^{1- δ})$ error term. We also prove new lower bounds for restricted type difference sets and asymmetric sumsets in $\mathbb{R}^d$.
Publié le : 
@article{DAS_2022_a5,
     author = {Akshat Mudgal},
     title = {New lower bounds for cardinalities of higher dimensional difference sets and sumsets},
     journal = {Discrete analysis},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2022_a5/}
}
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AU  - Akshat Mudgal
TI  - New lower bounds for cardinalities of higher dimensional difference sets and sumsets
JO  - Discrete analysis
PY  - 2022
UR  - http://geodesic.mathdoc.fr/item/DAS_2022_a5/
LA  - en
ID  - DAS_2022_a5
ER  - 
%0 Journal Article
%A Akshat Mudgal
%T New lower bounds for cardinalities of higher dimensional difference sets and sumsets
%J Discrete analysis
%D 2022
%U http://geodesic.mathdoc.fr/item/DAS_2022_a5/
%G en
%F DAS_2022_a5
Akshat Mudgal. New lower bounds for cardinalities of higher dimensional difference sets and sumsets. Discrete analysis (2022). http://geodesic.mathdoc.fr/item/DAS_2022_a5/