Maximal upper asymptotic density of sets of integers with missing differences from a given set

Ram Krishna Pandey

Mathematica Bohemica (2015)

  • Volume: 140, Issue: 1, page 53-69
  • ISSN: 0862-7959

Abstract

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Let M be a given nonempty set of positive integers and S any set of nonnegative integers. Let δ ¯ ( S ) denote the upper asymptotic density of S . We consider the problem of finding μ ( M ) : = sup S δ ¯ ( S ) , where the supremum is taken over all sets S satisfying that for each a , b ∈ S , a - b ∉ M . In this paper we discuss the values and bounds of μ ( M ) where M = { a , b , a + n b } for all even integers and for all sufficiently large odd integers n with a < b and gcd ( a , b ) = 1 .

How to cite

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Pandey, Ram Krishna. "Maximal upper asymptotic density of sets of integers with missing differences from a given set." Mathematica Bohemica 140.1 (2015): 53-69. <http://eudml.org/doc/269875>.

@article{Pandey2015,
abstract = {Let $M$ be a given nonempty set of positive integers and $S$ any set of nonnegative integers. Let $\overline\{\delta \}(S)$ denote the upper asymptotic density of $S$. We consider the problem of finding \[\mu (M):=\sup \_\{S\}\overline\{\delta \}(S),\] where the supremum is taken over all sets $S$ satisfying that for each $a,b\in S$, $a-b \notin M.$ In this paper we discuss the values and bounds of $\mu (M)$ where $M = \lbrace a,b,a+nb\rbrace $ for all even integers and for all sufficiently large odd integers $n$ with $a<b$ and $\gcd (a,b)=1.$},
author = {Pandey, Ram Krishna},
journal = {Mathematica Bohemica},
keywords = {upper asymptotic density; maximal density},
language = {eng},
number = {1},
pages = {53-69},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Maximal upper asymptotic density of sets of integers with missing differences from a given set},
url = {http://eudml.org/doc/269875},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Pandey, Ram Krishna
TI - Maximal upper asymptotic density of sets of integers with missing differences from a given set
JO - Mathematica Bohemica
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 140
IS - 1
SP - 53
EP - 69
AB - Let $M$ be a given nonempty set of positive integers and $S$ any set of nonnegative integers. Let $\overline{\delta }(S)$ denote the upper asymptotic density of $S$. We consider the problem of finding \[\mu (M):=\sup _{S}\overline{\delta }(S),\] where the supremum is taken over all sets $S$ satisfying that for each $a,b\in S$, $a-b \notin M.$ In this paper we discuss the values and bounds of $\mu (M)$ where $M = \lbrace a,b,a+nb\rbrace $ for all even integers and for all sufficiently large odd integers $n$ with $a<b$ and $\gcd (a,b)=1.$
LA - eng
KW - upper asymptotic density; maximal density
UR - http://eudml.org/doc/269875
ER -

References

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  1. Cantor, D. G., Gordon, B., 10.1016/0097-3165(73)90003-4, J. Comb. Theory, Ser. A 14 281-287 (1973). (1973) Zbl0277.10043MR0371849DOI10.1016/0097-3165(73)90003-4
  2. Gupta, S., 10.1006/jcta.1999.3003, J. Comb. Theory, Ser. A 89 Article ID jcta.1999.3003, 55-69 (2000). (2000) Zbl0956.11006MR1736136DOI10.1006/jcta.1999.3003
  3. Gupta, S., Tripathi, A., 10.4064/aa-89-3-255-257, Acta Arith. 89 255-257 (1999). (1999) Zbl0932.11009MR1691854DOI10.4064/aa-89-3-255-257
  4. Haralambis, N. M., 10.1016/0097-3165(77)90076-0, J. Comb. Theory, Ser. A 23 22-33 (1977). (1977) Zbl0359.10047MR0453689DOI10.1016/0097-3165(77)90076-0
  5. Liu, D. D., Zhu, X., 10.1002/jgt.20020, J. Graph Theory 47 129-146 (2004). (2004) MR2082743DOI10.1002/jgt.20020
  6. Liu, D. D., Zhu, X., 10.1016/j.ejc.2007.09.007, Eur. J. Comb. 29 1733-1743 (2008). (2008) Zbl1246.05058MR2431764DOI10.1016/j.ejc.2007.09.007
  7. Motzkin, T. S., Unpublished problem collection, . 
  8. Pandey, R. K., Tripathi, A., A note on a problem of Motzkin regarding density of integral sets with missing differences, J. Integer Seq. (electronic only) 14 Article 11.6.3, 8 pages (2011). (2011) MR2811219
  9. Pandey, R. K., Tripathi, A., 10.1016/j.jnt.2010.09.013, J. Number Theory 131 634-647 (2011). (2011) Zbl1229.11044MR2753268DOI10.1016/j.jnt.2010.09.013
  10. Pandey, R. K., Tripathi, A., On the density of integral sets with missing differences, {Combinatorial Number Theory. Proceedings of the 3rd `Integers Conference 2007', Carrollton, USA, 2007} B. Landman et al. Walter de Gruyter, Berlin 157-169 (2009). (2009) Zbl1218.11009MR2521959
  11. Rabinowitz, J. H., Proulx, V. K., 10.1137/0606050, SIAM J. Algebraic Discrete Methods 6 507-518 (1985). (1985) Zbl0579.05058MR0791178DOI10.1137/0606050

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