[ptspin]

I am certainly no expert on the subject but I would suggest the following points as answers to your question.

1) Most (that I have heard of) plans for Land Value Tax do still include a smaller portion of property taxes. This does not completely solve the problem you pose but does ameliorate it a little.

2) Some of those infrastructure things are paid for via utility payments on utilities used. The 100 families in the apartment building will use much more water than a single family home. Some part of that money should be budgeted and used for improving the utilities available. That doesn't account for things like parks, fire department, etc. but does take up some of the slack.

3) lastly and most importantly those homes will be paying more than their fair share on purpose! If they live in a neighborhood where it is economical to build 100 unit apartment buildings then a lot with just a single-family home is being wildly underutilized. The higher tax costs will encourage owners to redevelop those properties into apartments or other higher value uses to make efficient use of that land. This is the entire point of Land Value Tax! It makes it in the interest of land owners to upgrade and improve their property (whereas traditional property taxes have a negative incentive because building a new building will increase your taxes).

edit: formatting