Another way to understand the dimension of mass:

 

 

If the distance decreases by half:

A mass M situated at a distance R from the Earth undergoes a force of attraction F where F = K.m.Mt / R²
Its acceleration towards the Earth is a = K.Mt / R² (acceleration = force divided by mass = F / m)
Initially stationary, at the end of time T this mass m will have covered a distance of d = a.T²/2 = (K.Mt / R²).(T²/2)

d = (K Mt / R²) (T²/2) or d R²= (K Mt ) (T²/2)         (1)

If we decrease the distances (R and d) by half (as shown in the figure on the right), as if space was shrinking, d R²= (K Mt ) (T²/2) becomes

(d/2) (R²/4) = (1/8) d.R² = (K.Mt) (T²/2)                  (2)

So that equation (2) is equivalent to equation (1) , or in other words, so that in scenario (2) the law of attraction of masses is the same as in scenario (1), it will be necessary that either:

Therefore, in order to conserve the same value of K constant (Newton constant) and the law of attraction of masses in a world "shrinked by half", it is necessary to divide the masses by 8.
When reducing all distances in a space with y factor, the masses will be reduced by a factor of y³, which allows us to conclude that mass will depend on space and will encompass in its dimensions the factor L³.

 

In a more simplified manner, we can use the following reasoning:

If the Earth's mass, Mt, does not change when all the distances are reduced by half, mass m, being two times closer to the Earth, should be subject to a force 4 times stronger and therefore have its acceleration increase by 4. As a result, the distance covered, d, will also be 4 times longer. In reality, instead of being 4 times longer, distance d decreases by half. There is therefore a distance of factor 8, or 2³.
To conserve the law of attraction of masses (as well as the Newton constant) it will be necessary to decrease the distances by half and the masses by a factor 8.
That allows us to conclude that mass will depend on space and will encompass in its dimensions the factor L³.

 

If time decreases by half (time becomes slower)

A mass m situated at a distance R from Earth is subject to an attraction force F where: F = K.m.Mt / R²
Its acceleration towards Earth is a = K.Mt / R² (force divided by mass = acceleration)

At the end of a period of time T this mass m will have covered a distance d = a.T²/2 = (K.Mt / R²).(T²/2)
d = (K.Mt / R²).(T²/2) or d = (1/2).(K.Mt / R²).T²   (3)

If this mass covers distance d in half the time (T is replaced by T/2 )  (3)  becomes:
d = (1/2).(K.Mt / R²).(T/2)² or d = (1/2).(1/4).(K.Mt / R²).T²   (4)

Comparing (3) and (4) :

d = (1/2).(K.Mt / R²).T²             (3)
d = (1/2).(1/4).(K.Mt / R²).T²   (4)

So that equation (3) is equivalent to equation (4) it will be necessary that either:

That allows us to conclude that mass will depend on time and will encompass the factor T-2 in its dimensions.

 

In a more simplified manner..

The same reasoning goes for Time, if time decreases by half (becomes slower)
To accelerate and cover a same distance d in half the time, it will be necessary to accelerate by 4 times, to apply a force 4 times stronger, and use a mass 4 times bigger to produce this same attraction force.
To conserve the value of the Newton constant, the mass should increase by a factor of 4 when time becomes two times slower.

We can observe that in the scenario where distances vary but time does not, densities (mass by unit of volume) will be conserve