No, the density of far distant parts of the universe is not expected to be greater than nearby parts due to the expansion of the universe. In fact, the opposite is true.
The expansion of the universe, as described by the theory of cosmic expansion, does not involve the creation or destruction of matter. Instead, it refers to the expansion of space itself between galaxies and clusters of galaxies. As the universe expands, the distances between objects increase, but the overall density of matter remains relatively constant.
On large scales, such as the scale of galaxy clusters or superclusters, the matter is distributed relatively uniformly. Therefore, the average density of matter in the universe remains fairly constant regardless of the expansion. This means that the density of distant parts of the universe is not expected to be significantly different from nearby parts.
However, it's worth noting that on very large scales, the expansion of the universe can affect the observable density of objects due to the redshift of light. The expansion causes the wavelength of light to stretch, resulting in a redshift. This redshift can make distant objects appear dimmer and less dense compared to similar objects that are closer to us. But this effect is due to the expansion of space itself, not because the distant parts of the universe have a higher density.