OFFSET
1,3
COMMENTS
a(n) is the minimum diameter of a 4-Golomb Ruler.
a(n) >= (n^2-2n^{3/2}+n)/4+sqrt(n)-1, by a version of the Erdos-Turan argument for Sidon sets.
a(q) <= (q^2-1)/4 - 1, if q is a prime power, proved by Caicedo, Martos and Trujillo (2015).
a(n) ~ n^2/4, proved by Caicedo, Martos and Trujillo (2015).
a(n) >= (n^2 - (2-e)n^{3/2})/4 for some e>0 and sufficiently large n, proved by Balogh, Füredi and Roy (2023).
a(n) >= (n^2 - (2-2^{-16})n^{3/2})/4 for sufficiently large n, proved by Carter, Hunter and O'Bryant (2025).
REFERENCES
M. D. Atkinson and A. Hassenklover, Sets of integers with distinct differences, Sch Comput. Sci., Carleton Univ., Ottawa, Ont., Canada, Rep. SCS-TR-63, Aug 1984.
M. D. Atkinson, N. Santoro, and J. Urrutia, Integer sets with distinct sums and differences and carrier frequency assignments for nonlinear repeaters, IEEE Transactions on Communications, Vol. Com-34, No. 6, June 1986.
J. Balogh, Z. Füredi and S. Roy, An upper bound on the size of Sidon sets, Amer. Math. Monthly 130 (2023), no. 5, 437-445.
Y. Caicedo, C. Martos, and C. Trujillo, g-Golomb rulers, Rev. Integr. Mat. 33 (2015), no. 2, 161-172.
D. Carter, Z. Hunter and K. O’Bryant, On the diameter of finite Sidon sets, Acta Math. 175 (2025), no. 1, 108-126.
C. Martos, D. Daza and C. Trujillo, Near-optimal g-Golomb rulers, IEEE Access 9 (2021), 65482-65489.
LINKS
Aditya Gupta, Optimal Diameters of High Multiplicity g-Golomb Rulers, arXiv:2605.14229 [math.CO], 2026. See p. 2.
Kevin O'Bryant, Code for generating g-Golomb Rulers.
EXAMPLE
For n=18: {0, 1, 2, 4, 8, 13, 18, 22, 24, 29, 32, 37, 41, 44, 47, 53, 54, 55}.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Kevin O'Bryant, Jan 15 2026
EXTENSIONS
a(17)-a(18) from Aditya A Gupta, Apr 05 2026
STATUS
approved