Quantitative stability for sumsets in R
نویسندگان
چکیده
Given a measurable set A ⊂ R of positive measure, it is not difficult to show that |A+A| = |2A| if and only if A is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If (|A + A| − |2A|)/|A| is small, is A close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between A and its convex hull in terms of (|A+A| − |2A|)/|A|.
منابع مشابه
Restricted Sumsets in Finite Vector Spaces : the Case
We determine the sharp lower bound for the cardinality of the restricted sumset A+′B = {a + b | a ∈ A, b ∈ B, a 6= b}, where A,B run over all subsets of size r = s = 1 + 3 in a vector space over F3. This solves a conjecture stated in an earlier paper of ours on sumsets and restricted sumsets in finite vector spaces. The analogous problem for an arbitrary prime p remains open. However, we do pro...
متن کاملThe additive structure of the squares inside rings
When defining the amount of additive structure on a set it is often convenient to consider certain sumsets; Calculating the cardinality of these sumsets can elucidate the set’s underlying structure. We begin by investigating finite sets of perfect squares and associated sumsets. We reveal how arithmetic progressions efficiently reduce the cardinality of sumsets and provide estimates for the min...
متن کاملSumsets in dihedral groups
Let Dn be the dihedral group of order 2n. For all integers r, s such that 1 ≤ r, s ≤ 2n, we give an explicit upper bound for the minimal size μDn (r, s) = min |A · B| of sumsets (product sets) A · B, where A and B range over all subsets of Dn of cardinality r and s respectively. It is shown by construction that μDn (r, s) is bounded above by the known value of μG (r, s), where G is any abelian ...
متن کاملA Quantitative Result on Diophantine Approximation for Intersective Polynomials
In this short note, we closely follow the approach of Green and Tao to extend the best known bound for recurrence modulo 1 from squares to the largest possible class of polynomials. The paper concludes with a brief discussion of a consequence of this result for polynomial structures in sumsets and limitations of the method.
متن کاملThe number of sumsets in a finite field
We prove that there are 2p/2+o(p) distinct sumsets A + B in Fp where |A|, |B| → ∞ as p →∞.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013