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- W1775682007 abstract "In this paper we investigate some new problems in additive combinatorics. Our problems mainly involve permutations (or circular permutations) $n$ distinct numbers (or elements of an additive abelian group) $a_1,ldots,a_n$ with adjacent sums $a_i+a_{i+1}$ (or differences $a_i-a_{i+1}$) pairwise distinct. For an odd prime power $q=2n+1>13$ with $qnot=25$, we show that there is a circular permutation $(a_1,ldots,a_n)$ of the elements of $S={a^2: ainmathbb F_qsetminus{0}}$ such that ${a_1+a_2,ldots,a_{n-1}+a_n,a_n+a_1}=S$, where $mathbb F_q$ denotes the field of order $q$. For any finite subset $A$ of an additive torsion-free abelian group $G$ with $|A|=n>3$, we prove that there is a numbering $a_1,ldots,a_n$ of the elements of $A$ such that $$a_1+2a_2, a_2+2a_3, ldots, a_{n-1}+2a_n, a_n+2a_1$$ are pairwise distinct. We also pose 30 open conjectures for further research." @default.
- W1775682007 created "2016-06-24" @default.
- W1775682007 creator A5075574908 @default.
- W1775682007 date "2013-09-06" @default.
- W1775682007 modified "2023-09-27" @default.
- W1775682007 title "Some new problems in additive combinatorics" @default.
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