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DIRECTIONS SETS: A GENERALISATION OF RATIO SETS

Abstract

For every integer $k\geq 2$ and every $A\subseteq \mathbb{N}$ , we define the $k$ -directions sets of $A$ as $D^{k}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{k}\}$ and $D^{\text{}\underline{k}}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{\text{}\underline{k}}\}$ , where $\Vert \cdot \Vert$ is the Euclidean norm and $A^{\text{}\underline{k}}:=\{\boldsymbol{a}\in A^{k}:a_{i}\neq a_{j}\text{ for all }i\neq j\}$ . Via an appropriate homeomorphism, $D^{k}(A)$ is a generalisation of the ratio set $R(A):=\{a/b:a,b\in A\}$ . We study $D^{k}(A)$ and $D^{\text{}\underline{k}}(A)$ as subspaces of $S^{k-1}:=\{\boldsymbol{x}\in [0,1]^{k}:\Vert \boldsymbol{x}\Vert =1\}$ . In particular, generalising a result of Bukor and Tóth, we provide a characterisation of the sets $X\subseteq S^{k-1}$ such that there exists $A\subseteq \mathbb{N}$ satisfying $D^{\text{}\underline{k}}(A)^{\prime }=X$ , where $Y^{\prime }$ denotes the set of accumulation points of $Y$ . Moreover, we provide a simple sufficient condition for $D^{k}(A)$ to be dense in $S^{k-1}$ . We conclude with questions for further research.

Footnotes

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P. Leonetti is supported by the Austrian Science Fund (FWF), project F5512-N26; C. Sanna is supported by a postdoctoral fellowship of INdAM and is a member of the INdAM group GNSAGA.

References

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[1] Brown, B., Dairyko, M., Garcia, S. R., Lutz, B. and Someck, M., ‘Four quotient set gems’, Amer. Math. Monthly 121(7) (2014), 590599.
[2] Bukor, J. and Tóth, J. T., ‘On accumulation points of ratio sets of positive integers’, Amer. Math. Monthly 103(4) (1996), 502504.
[3] Bukor, J., Erdős, P., Šalát, T. and Tóth, J. T., ‘Remarks on the (R)-density of sets of numbers. II’, Math. Slovaca 47(5) (1997), 517526.
[4] Bukor, J., Šalát, T. and Tóth, J. T., ‘Remarks on R-density of sets of numbers’, Tatra Mt. Math. Publ. 11 (1997), 159165.
[5] Chattopadhyay, J., Roy, B. and Sarkar, S., ‘On fractionally dense sets’, Rocky Mountain J. Math. (to appear), https://projecteuclid.org:443/euclid.rmjm/1539914457.
[6] Donnay, C., Garcia, S. R. and Rouse, J., ‘ p-adic quotient sets II: Quadratic forms’, J. Number Theory 201 (2019), 2339.
[7] Garcia, S. R., ‘Quotients of Gaussian primes’, Amer. Math. Monthly 120(9) (2013), 851853.
[8] Garcia, S. R. and Luca, F., ‘Quotients of Fibonacci numbers’, Amer. Math. Monthly 123(10) (2016), 10391044.
[9] Garcia, S. R., Hong, Y. X., Luca, F., Pinsker, E., Sanna, C., Schechter, E. and Starr, A., ‘ p-adic quotient sets’, Acta Arith. 179(2) (2017), 163184.
[10] Garcia, S. R., Selhorst-Jones, V., Poore, D. E. and Simon, N., ‘Quotient sets and Diophantine equations’, Amer. Math. Monthly 118(8) (2011), 704711.
[11] Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, 6th edn (Oxford University Press, Oxford, 2008), revised by D. R. Heath-Brown and J. H. Silverman, with a foreword by Andrew Wiles.
[12] Hedman, S. and Rose, D., ‘Light subsets of ℕ with dense quotient sets’, Amer. Math. Monthly 116(7) (2009), 635641.
[13] Hobby, D. and Silberger, D. M., ‘Quotients of primes’, Amer. Math. Monthly 100(1) (1993), 5052.
[14] Miska, P. and Sanna, C., ‘ p-adic denseness of members of partitions of ℕ and their ratio sets’, Bull. Malays. Math. Sci. Soc. (2) (to appear), https://doi.org/10.1007/s40840-019-00728-6.
[15] Miska, P., Murru, N. and Sanna, C., ‘On the p-adic denseness of the quotient set of a polynomial image’, J. Number Theory 197 (2019), 218227.
[16] Pan, M. and Zhang, W., ‘Quotients of Hurwitz primes’, Preprint, 2019, arXiv:1904.08002.
[17] Sanna, C., ‘The quotient set of k-generalised Fibonacci numbers is dense in ℚp ’, Bull. Aust. Math. Soc. 96(1) (2017), 2429.
[18] Sittinger, B. D., ‘Quotients of primes in an algebraic number ring’, Notes Number Theory Discrete Math. 24(2) (2018), 5562.
[19] Starni, P., ‘Answers to two questions concerning quotients of primes’, Amer. Math. Monthly 102(4) (1995), 347349.
[20] Strauch, O. and Tóth, J. T., ‘Asymptotic density of AN and density of the ratio set R (A)’, Acta Arith. 87(1) (1998), 6778.
[21] Strauch, O. and Tóth, J. T., ‘Corrigendum to Theorem 5 of the paper: “Asymptotic density of A ⊂ℕ and density of the ratio set R (A)”’, Acta Arith. 103(2) (2002), 191200; Acta Arith. 87(1) (1998), 67–78.
[22] Šalát, T., ‘On ratio sets of sets of natural numbers’, Acta Arith. 15 (1968/1969), 273278.
[23] Šalát, T., ‘Corrigendum to the paper “On ratio sets of sets of natural numbers”’, Acta Arith. 16 (1969/1970), 103.
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DIRECTIONS SETS: A GENERALISATION OF RATIO SETS

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