Use of Air Voids with a nuclear densitometer

The gauge will give a reading on the bulk density, moisture content (and dry density). The gauge should be calibrated against the soil, so the readings can be corrected.
The results will give you the dry density and moisture content can then be used to calculate air void content, for which you would also need to know the particle density.
Adopting the moisture content as a percentage, of say, 15%, a dry density of 1.65 Mg/m³ and a particle density of 2.65 Mg/m³, the calculation would be:
The air void content = {1.00m³ -(dry density/particle density)-(dry density x (moisture content / 100))} x 100(%)
Air void = {1.00m³ -(1.65/2.65)-(1.65x(15/100))} x 100(%)
Air void = {1.00m³ - 0.623 m³ - 0.248 m³} = 0.130m³ x 100(%) = 13.0%

If you are using the gauge to calculate the air void content direct, please be aware that the readings are not corrected to the soil under teat and that you will have to ensure you have entered the correct particle density for the soil under analysis.

The use of air void contents is a guide to an end product assessment of fill, best suited for cohesive fill material. In the UK, the code of practice for earthworks would be a good starting point, BS 6031:2009, and across Europe including the UK, you also have BS EN 16907-1 TO 6: 2018, which takes through soil classification all the way to QA and verification.