{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T00:31:34Z","timestamp":1768869094067,"version":"3.49.0"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T00:00:00Z","timestamp":1759276800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T00:00:00Z","timestamp":1762905600000},"content-version":"vor","delay-in-days":42,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Acta Math. Hungar."],"published-print":{"date-parts":[[2025,10]]},"abstract":"Abstract<\/jats:title>\n \n Given a set\n \n \n $$X\\subseteq\\mathbb{R}^2$$<\/jats:tex-math>\n \n \n X<\/mml:mi>\n \u2286<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of\n \n \n $$n$$<\/jats:tex-math>\n \n n<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n points and a distance\n \n \n $$d>0$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , the multiplicity of\n \n \n $$d$$<\/jats:tex-math>\n \n d<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is the\nnumber of times the distance\n \n \n $$d$$<\/jats:tex-math>\n \n d<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n appears between points in\n \n \n $$X$$<\/jats:tex-math>\n \n X<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Let\n \n \n $$a_1(X) \\geq a_2(X) \\geq \\cdots \\geq a_m(X)$$<\/jats:tex-math>\n \n \n \n a<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2265<\/mml:mo>\n \n a<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2265<\/mml:mo>\n \u22ef<\/mml:mo>\n \u2265<\/mml:mo>\n \n a<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n denote the multiplicities of the\n \n \n $$m$$<\/jats:tex-math>\n \n m<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n distances determined by\n \n \n $$X$$<\/jats:tex-math>\n \n X<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and let\n \n \n $$a(X)=(a_1(X),\\dots,a_m(X))$$<\/jats:tex-math>\n \n \n a<\/mml:mi>\n \n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n (<\/mml:mo>\n \n a<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n \n a<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In this paper, we study several\nquestions from Erd\u0151s\u2019s time regarding distance multiplicities. Among other results, we show that:\n \n \n \n (1) If\n \n \n $$X$$<\/jats:tex-math>\n \n X<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is convex or \u201cnot too convex\u201d, then there exists a distance other than the diameter\n that has multiplicity at most\n \n \n $$n$$<\/jats:tex-math>\n \n n<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>\n <\/jats:list-item>\n \n \n (2) There exists a set\n \n \n $$X\\subseteq\\mathbb{R}^2$$<\/jats:tex-math>\n \n \n X<\/mml:mi>\n \u2286<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of\n \n \n $$n$$<\/jats:tex-math>\n \n n<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n points, such that many distances occur with high\n multiplicity. In particular, at least\n \n \n $$n^{\\Omega(1\/\\log\\log{n})}$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \n \u03a9<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n log<\/mml:mo>\n log<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n distances \n have superlinear multiplicity in\n \n \n $$n$$<\/jats:tex-math>\n \n n<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>\n <\/jats:list-item>\n \n \n (3) For any (not necessarily fixed) integer\n \n \n $$1\\leq k\\leq\\log{n}$$<\/jats:tex-math>\n \n \n 1<\/mml:mn>\n \u2264<\/mml:mo>\n k<\/mml:mi>\n \u2264<\/mml:mo>\n log<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ,\n there exists\n \n \n $$ {X\\subseteq\\mathbb{R}^2 } $$<\/jats:tex-math>\n \n \n X<\/mml:mi>\n \u2286<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of\n \n \n $$n$$<\/jats:tex-math>\n \n n<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n points, such that the difference between\n the\n \n \n $$k^{\\text{th}}$$<\/jats:tex-math>\n \n \n k<\/mml:mi>\n th<\/mml:mtext>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $$(k+1)^{\\text{th}}$$<\/jats:tex-math>\n \n \n \n (<\/mml:mo>\n k<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n th<\/mml:mtext>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n largest multiplicities is at least\n \n \n $$\\Omega(\\frac{n\\log{n}}{k})$$<\/jats:tex-math>\n \n \n \u03a9<\/mml:mi>\n (<\/mml:mo>\n \n \n n<\/mml:mi>\n log<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n k<\/mml:mi>\n <\/mml:mfrac>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . \n Moreover, the distances in\n \n \n $$X$$<\/jats:tex-math>\n \n X<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n with the largest\n \n \n $$k$$<\/jats:tex-math>\n \n k<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n multiplicities can be prescribed.\n <\/jats:p>\n <\/jats:list-item>\n \n \n (4) For every\n \n \n $$n\\in N$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , there exists\n \n \n $$X\\subseteq\\mathbb{R}^2$$<\/jats:tex-math>\n \n \n X<\/mml:mi>\n \u2286<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of\n \n \n $$n$$<\/jats:tex-math>\n \n n<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n points, not all collinear or cocircular, such that\n \n \n $$a(X)= (n-1,n-2,\\ldots,1)$$<\/jats:tex-math>\n \n \n a<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n (<\/mml:mo>\n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n n<\/mml:mi>\n -<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . There also exists\n \n \n $$X\\subseteq\\mathbb{R}^2$$<\/jats:tex-math>\n \n \n X<\/mml:mi>\n \u2286<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of\n \n \n $$n$$<\/jats:tex-math>\n \n n<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n points with pairwise distinct distance multiplicities and\n \n \n $$a(Y) \\neq (n-1,n-2,\\ldots,1)$$<\/jats:tex-math>\n \n \n a<\/mml:mi>\n (<\/mml:mo>\n Y<\/mml:mi>\n )<\/mml:mo>\n \u2260<\/mml:mo>\n (<\/mml:mo>\n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n n<\/mml:mi>\n -<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>\n <\/jats:list-item>\n <\/jats:list>\n <\/jats:p>","DOI":"10.1007\/s10474-025-01562-y","type":"journal-article","created":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T11:56:37Z","timestamp":1762948597000},"page":"231-245","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On multiplicities of interpoint distances"],"prefix":"10.1007","volume":"177","author":[{"given":"F. 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