Matches in SemOpenAlex for { <https://semopenalex.org/work/W2907315046> ?p ?o ?g. }
Showing items 1 to 63 of
63
with 100 items per page.
- W2907315046 abstract "A set $cal P$ of $n$ points in $R^d$ is separated if all distances of distinct points are at least~$1$. Then we may ask how many of these distances, with multiplicity, lie in an interval $[t, t + 1]$. The authors and J. Spencer proved that the maximum is $(n^2/2)(1 - 1/d) + O(1)$. The authors showed that for $d = 2$ and $cal P$ separated, the maximal number of distances, with multiplicity, in the union of $k$ unit intervals is $(n^2/2)$ $(1 - 1/(k + 1) + o(1))$. (In these papers the unit intervals could be replaced by intervals of length $text{const}_dcdot n^{1/d}$.) In this paper we show that for $k = 2$, and for any $n$, this maximal number is $(n^2/2)(1 - 1/m_{d - 1} + o(1))$, where $m_{d - 1}$ is the maximal size of a two-distance set in $R^{d - 1}$. (The value of $m_{d - 1}$ is known for $d - 1 leq 8$, and for each $d$ it lies in $left[left({datop 2}right), left({d + 1atop 2}right)right]$. For $d neq 4,5$ we can replace unit intervals by intervals of length $text{const}_d cdot n^{1/d}$, and the maximum is the respective Turan number, for $n geq n(d)$.) We also investigate a variant of this question, namely with $k$ intervals of the form $[t, t(1 + varepsilon)]$, for $varepsilon n(d, k)$. Here the maximal number of distances, with multiplicity, in the union of $k$ such intervals is the Turan number $T(n, (d + 1)^k + 1)$. Several of these results were announced earlier by Makai-Pach-Spencer." @default.
- W2907315046 created "2019-01-11" @default.
- W2907315046 creator A5008569200 @default.
- W2907315046 creator A5035271865 @default.
- W2907315046 creator A5074138092 @default.
- W2907315046 date "2019-01-04" @default.
- W2907315046 modified "2023-10-16" @default.
- W2907315046 title "Two nearly equal distances in $R^d$" @default.
- W2907315046 cites W2091514065 @default.
- W2907315046 cites W2330770891 @default.
- W2907315046 cites W2899286789 @default.
- W2907315046 hasPublicationYear "2019" @default.
- W2907315046 type Work @default.
- W2907315046 sameAs 2907315046 @default.
- W2907315046 citedByCount "1" @default.
- W2907315046 countsByYear W29073150462019 @default.
- W2907315046 crossrefType "posted-content" @default.
- W2907315046 hasAuthorship W2907315046A5008569200 @default.
- W2907315046 hasAuthorship W2907315046A5035271865 @default.
- W2907315046 hasAuthorship W2907315046A5074138092 @default.
- W2907315046 hasConcept C114614502 @default.
- W2907315046 hasConcept C121332964 @default.
- W2907315046 hasConcept C122637931 @default.
- W2907315046 hasConcept C145420912 @default.
- W2907315046 hasConcept C156004811 @default.
- W2907315046 hasConcept C2524010 @default.
- W2907315046 hasConcept C2778067643 @default.
- W2907315046 hasConcept C33923547 @default.
- W2907315046 hasConceptScore W2907315046C114614502 @default.
- W2907315046 hasConceptScore W2907315046C121332964 @default.
- W2907315046 hasConceptScore W2907315046C122637931 @default.
- W2907315046 hasConceptScore W2907315046C145420912 @default.
- W2907315046 hasConceptScore W2907315046C156004811 @default.
- W2907315046 hasConceptScore W2907315046C2524010 @default.
- W2907315046 hasConceptScore W2907315046C2778067643 @default.
- W2907315046 hasConceptScore W2907315046C33923547 @default.
- W2907315046 hasLocation W29073150461 @default.
- W2907315046 hasOpenAccess W2907315046 @default.
- W2907315046 hasPrimaryLocation W29073150461 @default.
- W2907315046 hasRelatedWork W109072738 @default.
- W2907315046 hasRelatedWork W1914119915 @default.
- W2907315046 hasRelatedWork W2089817321 @default.
- W2907315046 hasRelatedWork W2094056166 @default.
- W2907315046 hasRelatedWork W2107079802 @default.
- W2907315046 hasRelatedWork W2149119567 @default.
- W2907315046 hasRelatedWork W2270191689 @default.
- W2907315046 hasRelatedWork W2336392368 @default.
- W2907315046 hasRelatedWork W2491173871 @default.
- W2907315046 hasRelatedWork W2766781494 @default.
- W2907315046 hasRelatedWork W2802650376 @default.
- W2907315046 hasRelatedWork W2905531951 @default.
- W2907315046 hasRelatedWork W2948862220 @default.
- W2907315046 hasRelatedWork W2951821892 @default.
- W2907315046 hasRelatedWork W2964187150 @default.
- W2907315046 hasRelatedWork W2990284752 @default.
- W2907315046 hasRelatedWork W2992211694 @default.
- W2907315046 hasRelatedWork W3118054965 @default.
- W2907315046 hasRelatedWork W3127350324 @default.
- W2907315046 hasRelatedWork W3174942294 @default.
- W2907315046 isParatext "false" @default.
- W2907315046 isRetracted "false" @default.
- W2907315046 magId "2907315046" @default.
- W2907315046 workType "article" @default.