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Results tagged with additive-combinatorics
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user 168142
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.
2
votes
1
answer
396
views
Sets with certain property concerning density of sumsets
I am working with subsets of $[n]$ of the form $(A+B)\cap A$, where $A+B$ is a sumset. Namely, I am interested if there are nonempty sets $B$ such that whenever $A$ covers a positive proportion of $[n …
- 114k
9
votes
1
answer
256
views
Arithmetic progressions in inverse image of totient function
I noticed on the OEIS that there are various sequences (e.g. A050515-A050520) that describe arithmetic progressions whose totients are all equal. For example, we have
$$\varphi(\{1,2\}) = 1$$
$$\varph …
- 3,320
7
votes
1
answer
263
views
Analogue to Szemerédi's theorem for non-monotone sequences
Szemerédi's theorem states that a strictly increasing sequence of positive integers $a_0, a_1, \ldots$ whose range has positive density contains arbitrarily long arithmetic progressions (as subsequenc …
- 807
2
votes
Accepted
Analogue to Szemerédi's theorem for non-monotone sequences
It appears the statement is false. The paper "On permutations containing no long arithmetic progressions," by Davis, Entringer, Graham, and Simmons [Acta Arithmetica 34 (1977)] exhibits a permutation …
- 807
10
votes
2
answers
921
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Converse to Erdős' conjecture on arithmetic progressions
I apologise in advance if this has been asked here before. I did a search and did not find anything obvious. Erdős' conjecture states that if $A\subseteq {\bf N}$ is such that $\sum_{n\in A} n^{-1}$ d …
- 105k