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In this paper, we study the quantity $$\\begin{aligned} M_k(d) = \\lim _{\\varepsilon \\rightarrow 0}\\max {\\{|S|:S\\,\\text { is an}\\, \\varepsilon \\text {-nearly}\\, k\\text {-distance set in}\\,{{\\mathbb {R}}}^d\\}} \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \n M<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n lim<\/mml:mo>\n \n \u03b5<\/mml:mi>\n \u2192<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:munder>\n max<\/mml:mo>\n \n {<\/mml:mo>\n |<\/mml:mo>\n S<\/mml:mi>\n |<\/mml:mo>\n :<\/mml:mo>\n S<\/mml:mi>\n \n \n is an<\/mml:mtext>\n \n \u03b5<\/mml:mi>\n -nearly<\/mml:mtext>\n \n k<\/mml:mi>\n -distance set in<\/mml:mtext>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula>and its relation to the classical quantity $$m_k(d)$$<\/jats:tex-math>\n \n \n m<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>: the size of the largest k<\/jats:italic>-distance set in\u00a0$${{\\mathbb {R}}}^d$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We obtain that $$M_k(d)=m_k(d)$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n m<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for $$k=2,3$$<\/jats:tex-math>\n \n k<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, as well as for any fixed\u00a0k<\/jats:italic>, provided that d<\/jats:italic> is sufficiently large. The last result answers a question, proposed by Erd\u0151s, Makai, and Pach. We also address a closely related Tur\u00e1n-type problem, studied by Erd\u0151s, Makai, Pach, and Spencer in the 90s: given n<\/jats:italic> points in\u00a0$${{\\mathbb {R}}}^d$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, how many pairs of them form a distance that belongs to $$[t_1,t_1+1]\\cup \\cdots \\cup [t_k,t_k+1]$$<\/jats:tex-math>\n \n \n [<\/mml:mo>\n \n t<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n t<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n \u222a<\/mml:mo>\n \u22ef<\/mml:mo>\n \u222a<\/mml:mo>\n \n [<\/mml:mo>\n \n t<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n ,<\/mml:mo>\n \n t<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n +<\/mml:mo>\n 1<\/mml:mn>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where $$t_1,\\dots ,t_k$$<\/jats:tex-math>\n \n \n t<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n \n t<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are fixed and any two points in the set are at distance at least 1\u00a0apart? We establish the connection between this quantity and a quantity closely related to $$M_k(d-1)$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, as well as obtain an exact answer for the same ranges k<\/jats:italic>,\u00a0d<\/jats:italic> as above.<\/jats:p>","DOI":"10.1007\/s00454-023-00489-x","type":"journal-article","created":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T18:42:34Z","timestamp":1686076954000},"page":"455-494","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Nearly k-Distance Sets"],"prefix":"10.1007","volume":"70","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4939-4835","authenticated-orcid":false,"given":"N\u00f3ra","family":"Frankl","sequence":"first","affiliation":[]},{"given":"Andrey","family":"Kupavskii","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,6]]},"reference":[{"issue":"2","key":"489_CR1","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/BF02579288","volume":"3","author":"E Bannai","year":"1983","unstructured":"Bannai, E., Bannai, E., Stanton, D.: An upper bound for the cardinality of an $$s$$-distance subset in real Euclidean space. 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