A370279 -id:A370279 - cp's OEIS frontend

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

Zachary DeStefano, Feb 13 2024

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.

			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.

Stack Exchange Dirichlet's Simultaneous Approximation Bound.

Cf. A370278, A370279.

A370278 Difference between the bound provided by Dirichlet's Simultaneous Approximation Theorem applied to Z_n (for d=3) and the best possible bound.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 0, 0, 2, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 2, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3

Offset: 2

Zachary DeStefano, Feb 13 2024

nonn

Indices where this sequence is 0 form the sequence A370277.

The indices of record high values form the sequence A370279.

			For n = 6, floor(k^(2/3)) = 3, but for all triples (a_1, a_2, a_3), there is a choice of p such that |p*a_1| mod 6, |p*a_2| mod 6, and |p*a_3| mod 6 are all smaller than or equal to 2.
For example, consider the triple (1, 2, 3), with p = 2; we have:
|2 * 1| mod 6 = 2, |2 * 2| mod 6 = 2, and |2 * 3| mod 6 = 0.
Note that there is no nonzero choice of p such that all values are smaller than 2 for this triple.

Cf. A370277, A370279.

cp's OEIS Frontend

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

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