Tight Acute Sets

Question

If the five 3D coordinates 003 044 330 404 443 are plotted, all the angles happen to be acute, so this is an acute set. It is believed that five points is the largest acute set in 3 dimensions. Also, all the points are on the integer grid, and the largest-smallest grid values is 4-0 or 4, so this acute set has a tightness of 4.

In 4D, the acute set {{8,8,0,0}, {-8,-8,0,0}, {0,8,8,2}, {0,-8,-8,2}, {8,0,8,1}, {-8,0,-8,1}, {6,5,6,8}, {-6,-5,-6,8}} has a tightness of 16. It is believed that 8 points is the largest acute set in 4D.

For real starting values I'm using Viktor Harangi's Acute sets in Euclidean spaces.

In 5D, the largest known acute set has 12 points. A tightness of 228 comes from
{{114,0,0,0,1}, {-114,0,0,0,1}, {0,114,0,0,2}, {0,-114,0,0,2},
{0,0,114,0,3}, {0,0,-114,0,3}, {0,0,0,114,0}, {0,0,0,-114,0},
{31,31,31,31,99}, {-31,-31,-31,-31,99}, {55,55,-55,-55,-30}, {-55,-55,55,55,-30}}

In 6D, the largest known acute set has 16 points.
In 7D, the largest known acute set has 24 points.
In 8D, the largest known acute set has 32 points.
In 9D, the largest known acute set has 48 points.
In 10D, the largest known acute set has 72 points.

What are the tightest presentations for maximal acute sets? Do I even have it right for 3D and 4D?

Answer 1

Recently, new results have been obtained for this problem.

First, D. Zakharov constructed an acute set in $\mathbb R^d$ of size at least $1.618^d$.

Thereafter Balázs Gerencsér and Viktor Harangi presented a simple construction of an acute set of size $2^{d−1}+1$ in $\mathbb R^d$ for any dimension $d$.

My examples for maximal accute sets:

Example for 4D-9 points:

(0, 1, 2, 0)  (0, 42, 1, 1)  (42, 1, 0, 2)  (41, 41, 0, 13) 
(44, 43, 42, 0)  (44, 2, 43, 1)  (2, 43, 44, 2)  (3, 3, 44, 13) 
(32, 32, 15, 44)

I think that this is not the optimal example, but almost.

Example for 5D-17 points (this example is not optimal, I'm sure :)

(6000, 6000, 6000, 6000, 3100000) 
(199999776, 100, 300, 240, 196000) 
(14, 199999968, 0, 2, 601998) 
(200000000, 199999986, 0, 0, 695818) 
(0, 0, 199999986, 0, 695832) 
(200000000, 2, 200000000, 12, 695840) 
(0, 199999997, 199999984, 0, 727498) 
(199999998, 200000000, 199999988, 0, 695858) 

(199994000, 199994000, 199994000, 199994000, 3100000) 
(224, 199999900, 199999700, 199999760, 196000) 
(199999986, 32, 200000000, 199999998, 601998) 
(0, 14, 200000000, 200000000, 695818) 
(200000000, 200000000, 14, 200000000, 695832) 
(0, 199999998, 0, 199999988, 695840) 
(200000000, 3, 16, 200000000, 727498) 
(2, 0, 12, 200000000, 695858) 

(120000000, 120000000, 120000000, 120000000, 200000000)

Upd. 6.10.2017 I improved my understanding and algorithms for 4D-9 points:

(1, 5, 5, 9)  (2, 22, 4, 1)  (20, 2, 2, 0)  (22, 20, 2, 2)  
(23, 19, 19, 9)  (22, 2, 20, 1)  (4, 22, 22, 0)  (2, 4, 22, 2)  
(13, 12, 12, 24)

Answer 2

I found a slightly smaller solution for 9 points in 4D:

(1 10 3 8)   (12 19 1 9) (1 15 18 7) (10 1 7 0)
(21 12 19 8) (10 3 21 9) (21 7 4 7)  (12 21 15 0)
(11 11 11 22)