Abstract:
For an infinite set $\mathscr A\subset\mathbb N$, let the function $\rho(x,y;\mathscr A)$, where $x\in\mathbb N$ and $y+1\in\mathbb N$, be defined as the length of a maximal interval $(\alpha,\beta)\subset(y,x+y)$ disjoint from $\mathscr A$. We study the function $$ \rho_*(x;\mathscr A)=\inf_{y+1\in\mathbb N}\rho(x,y;\mathscr A). $$ In particular, we determine the order of this function for the set $\mathscr A$ of square-free numbers.
Keywords:
set of positive integers, prime, square-free number.
This work was performed at Steklov International Mathematical Center of
Steklov Mathematical Institute of Russian Academy of Sciences and financially supported
by the Ministry of Science and Higher Education of the Russian Federation
(grant no. 075-15-2025-303).
Citation:
S. V. Konyagin, “On the Local Distribution of Elements of Subsets of the Set of Positive Integers”, Mat. Zametki, 118:4 (2025), 515–528; Math. Notes, 118:4 (2025), 752–763
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\by S.~V.~Konyagin
\paper On the Local Distribution of Elements of Subsets of the Set of Positive Integers
\jour Mat. Zametki
\yr 2025
\vol 118
\issue 4
\pages 515--528
\mathnet{http://mi.mathnet.ru/mzm14596}
\crossref{https://doi.org/10.4213/mzm14596}
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\transl
\jour Math. Notes
\yr 2025
\vol 118
\issue 4
\pages 752--763
\crossref{https://doi.org/10.1134/S0001434625605088}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-105025769584}
Citation in format AMSBIB
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