Almost all sets of nonnegative integers and their small perturbations are not sumsets

نویسندگان

چکیده

Fix α ∈ ( 0 , 1 / 3 stretchy="false">) \alpha \in (0,1/3) . We show that, from a topological point of view, almost all sets alttext="upper A subset-of-or-equal-to double-struck upper N"> A ⊆<!-- ⊆ <mml:mi mathvariant="double-struck">N encoding="application/x-tex">A\subseteq \mathbb {N} have the property if prime equals A"> class="MJX-variant" mathvariant="normal">′<!-- ′ </mml:msup> = encoding="application/x-tex">A^\prime =A for but alttext="o n Superscript alpha Baseline o n encoding="application/x-tex">o(n^{\alpha }) elements, then prime"> encoding="application/x-tex">A^\prime is not nontrivial sumset B plus C"> B + C encoding="application/x-tex">B+C In particular, encoding="application/x-tex">A are totally irreducible. addition, we prove that measure analogue holds with 1"> =1

برای دانلود باید عضویت طلایی داشته باشید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

When almost all sets are difference dominated

We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0, 1, ..., N}, chosen randomly according to a binomial model with parameter p(N), with N = o(p(N)). We show that the random subset is almost surely difference dominated, as N → ∞, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O’Bryant. Furthermore, we ex...

متن کامل

Almost all webs are not rank-perfect

Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [8,12] and claw-free graphs [7,8]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [9]. Ben Rebea conjectured a description for quasi-line graphs, see [12]; Chudnovsky and Seymour [2] v...

متن کامل

Arithmetic Progressions in Sets with Small Sumsets

We present an elementary proof that if A is a finite set of numbers, and the sumset A+G A is small, |A+G A| ≤ c|A|, along a dense graph G, then A contains k-term arithmetic progressions.

متن کامل

Large sets in finite fields are sumsets

For a prime p, a subset S of Zp is a sumset if S = A+A for some A ⊂ Zp. Let f(p) denote the maximum integer so that every subset S ⊂ Zp of size at least p− f(p) is a sumset. The question of determining or estimating f(p) was raised by Green. He showed that for all sufficiently large p, f(p) ≥ 19 log2 p and proved, with Gowers, that f(p) < cp 2/3 log p for some absolute constant c. Here we impro...

متن کامل

Sumsets of sparse sets

Let σ be a constant in the interval (0, 1), and let A be an infinite set of positive integers which contains at least c1x σ and at most c2x σ elements in the interval [1, x] for some constants c2 > c1 > 0 independent of x and each x ≥ x0. We prove that then the sumset A + A has more elements than A (counted up to x) by a factor c(σ) √ log x/ log log x for x large enough. An example showing that...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

ژورنال

عنوان ژورنال: Proceedings of the American Mathematical Society

سال: 2023

ISSN: ['2330-1511']

DOI: https://doi.org/10.1090/proc/16392