A370277 Numbers k with the property that Dirichlet's Simultaneous Approximation Theorem applied to Z_k is tight (for d = 3).

2, 4, 5, 7, 8, 10, 11, 14, 18, 26, 27, 30, 31, 63, 64, 68, 69, 70, 76, 124, 125, 130, 131, 132, 148, 215, 216, 222, 223, 224, 225, 234, 342, 343, 350, 351, 352, 353

Offset: 1

Author

Zachary DeStefano, Feb 13 2024

Keywords

• nonn
• more

Comments

Dirichlet's Simultaneous Approximation Theorem applied to Z_k states that for all a_1, a_2, ..., a_d, there exists a nonzero p such that |pa_i| <= k^(1 - 1/d) mod k.

For d = 3, the bound of floor(k^(2/3)) is tight only for specific values of k. That is to say, max_(a_1,a_2,a_3) min_p max_i |pa_i| = floor(k^(2/3)) only for specific values of k. These are those values.

This sequence consists of the indices of the zeros in A370278.

It appears that this sequence contains all integers k such that k or k+1 is a cube.

Examples

			For k = 14, floor(k^(2/3)) = 5. Given the triple (1, 3, 5), there is no choice of p such that |p| mod 14, |3p| mod 14, and |5p| mod 14 are all smaller than 5.
p = 1, 3, 5, 9, 11, and 13 results in a simultaneous minimum of 5.
		

Links

• Stack Exchange Dirichlet's Simultaneous Approximation Bound.

Crossrefs

Cf. A370278, A370279.