• Corpus ID: 239050550

# New lower bounds for cardinalities of higher dimensional difference sets and sumsets

@inproceedings{Mudgal2021NewLB,
title={New lower bounds for cardinalities of higher dimensional difference sets and sumsets},
author={Akshat Mudgal},
year={2021}
}
Let d ≥ 4 be a natural number and let A be a finite, non-empty subset of R such that A is not contained in a translate of a hyperplane. In this setting, we show that

## References

SHOWING 1-10 OF 11 REFERENCES
An Upper Bound for d-dimensional Difference Sets
Let be the maximal positive number for which the inequality holds for every finite set of affine dimension . What can one say about ? The exact value of is known only for d = 1, 2 and 3. It is shown
On Finite Difference Sets
We establish a lower estimate of the cardinality of finite difference sets in the three dimensional Euclidean space R3 by showing that ¦A - A¦ ≧ 4.5 ¦A¦ -9 for every three dimensional finite set A.
Difference sets in higher dimensions
• Akshat Mudgal
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2020
Abstract Let d ≥ 3 be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have \begin{equation*}
Sum of sets in several dimensions
• I. Ruzsa
• Mathematics, Computer Science
Comb.
• 1994
It is proved that the number of distinct vectors in the form {a+b∶a∈A, b∈B} is at leastn+dm−d(d+1)/2 and the exact bound for alln>2d is given.
Sums of Linear Transformations in Higher Dimensions
• Akshat Mudgal
• Mathematics
The Quarterly Journal of Mathematics
• 2019
In this paper, we prove the following two results. Let d be a natural number and q, s be co-prime integers such that 10 depending only on q, s and d such that for any finite subset A of ℝd that is
Properties of two-dimensional sets with small sumset
• Computer Science, Mathematics
J. Comb. Theory, Ser. A
• 2010
We give tight lower bounds on the cardinality of the sumset of two finite, nonempty subsets A,B@?R^2 in terms of the minimum number h"1(A,B) of parallel lines covering each of A and B. We show that,
A Brunn-Minkowski inequality for the integer lattice
• Mathematics
• 2001
A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the