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Results tagged with additive-combinatorics
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user 20598
Questions on the subject additive combinatorics, also known as arithmetic combinatorics, such as questions on: additive bases, sum sets, inverse sum set theorems, sets with small doubling, Sidon sets, Szemerédi's theorem and its ramifications, Gowers uniformity norms, etc. Often combined with the top-level tags nt.number-theory or co.combinatorics. Some additional tags are available for further specialization, including the tags sumsets and sidon-sets.
10
votes
2
answers
612
views
Maximize $f(0)+\cdots+f(n-1)$ subject to $f(x)f(y) + f(x+y) \leq 1$
Suppose $f:\mathbf{N} \to [0,1]$ satisfies $$f(x)f(y) + f(x+y)\leq 1\qquad(1)$$ for all $x,y$. Let $$d_n = \frac{1}{n} \sum_{x=0}^{n-1} f(x).$$ It is easy to prove that $$\limsup d_n \leq 1/\varphi,$$ …
- 9,720
4
votes
Maximize $f(0)+\cdots+f(n-1)$ subject to $f(x)f(y) + f(x+y) \leq 1$
Gerhard Paseman's solution is correct, and shows that $f \equiv 1/\phi$ is the unique maximizer of $d_n$ for each $n\geq 0$. Just reproducing here in as few words as I can, for the purpose of succinct …
- 9,720
6
votes
0
answers
151
views
A variant of the capset problem
Let $p > 2$ be a prime of bounded size.
Suppose $A$ is a subset of $G = \mathbf{F}_p^n$ with only degenerate solutions to
$$x + y = 2a,\\
x+z = 2b,\\
y + z = 2c,$$
where a solution is considered degen …
- 9,720
11
votes
Accepted
A sumset inequality
The proposed inequality is not true. I do not claim originality for this example: all I have done is take Seva's observation that the inequality would imply an improvement $|3A|\leq K^2|A|$ given $|2A …
- 9,720
9
votes
Accepted
Maximum density of sum-free sets with respect to Knuth's "addition"
Let $a +_K b$ denote Knuth addition. It is easy to check that $a +_K b \equiv a + b$ mod $2$ (in fact mod $4$), so the odd numbers are Knuth-sum-free. On the other hand, note that if $a +_K b = a +_K …
- 9,720
5
votes
Accepted
Partition regular systems: do they have solution in (very dense) set of integers?
This doesn't have anything to do with partition regularity: There is such a constant $C(A)<1$ provided only that there exists at least one solution to $Ax=0$ in positive integers.
Indeed suppose $x = …
- 9,720
6
votes
Accepted
The growth rate of a commutator set in a non-elementary group
You can take $\kappa(n) = n/2$ if $G$ is not virtually nilpotent of class $\le 2$.
Let $B_n = S^{\le n}$ and $C_n = \{[b_1, b_2] : b_1, b_2 \in B_n\}$. Suppose $|C_n| < n/2$. By pigeonhole there is so …
- 9,720
7
votes
Density version of the Erdős-Graham conjecture
Naturally, this was also considered by Erdös and Graham. Graham mentions at the top of page 10 here for instance: http://www.math.ucsd.edu/~ronspubs/13_03_Egyptian.pdf. I'm not aware of any progress.
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10
votes
What is the smallest cardinality of a self-linked set in a finite cyclic group?
The difference cover problem has been better studied in the context of $\mathbf{Z}$. Redei, Renyi, and others in the 40s asked for the size of the smallest set $A$ such that $A-A$ covers $\{1,2,\dots, …
- 9,720