I realise the headline flies in the face of the generally unquestioned idea that government spending gets the rich to shoulder more of the burden and benefits the poor, but I believe that the headline is true for a wide range of government expenditure.
Consider expenditure on transport infrastructure. If a new railway line or road is built to connect an outlying town to a city, it will generally make that area a more attractive commuter location and house prices will rise as a result. The same effect occurs around schools; the catchment area around a state school with a good reputation will generally have a premium on house prices. In effect, the state school system has a form of selection based on wealth, it is just less overt, as instead of the money being paid to the school, it is paid through the housing market. That isn't, however, the real regressive element. It is the way the housing market impacts on different groups which creates the major regressive effects.
As house prices increase, homeowners see an increase in their net worth; in contrast, people renting their home will find their rents increasing, resulting in them becoming worse off. So, as transport links, state schools, etc. improve in an area, people who own homes in an area will make a windfall gain, but renters will face greater costs. As renters will generally be poorer than homeowners, the overall effect is to benefit the wealthier while disadvantaging the poorer.
This leads me to draw two conclusions:
The first is that the situation could be improved by introducing an approach I've promoted regularly - Land Value Taxation. By introducing a land rent, in the form of a tax on the value of land titles, the regressive nature of public expenditure would be reduced, as those benefiting most from public expenditure would find themselves paying more tax in return.
The second conclusion may appear slightly more counter-intuitive - in order to reduce the regressive effects it creates, the government should provide far fewer services and leave the provision to a free market where the user pays the full cost directly. For example, instead of providing motorways directly, the government could leave the provision of motorways to the private sector or auction off the right to operate motorways to the highest bidder. That would increase the direct cost faced by drivers using the motorways, but massively reduce the premium on the price of houses which benefit from access to the motorways, reducing the windfall gains enjoyed by homeowners, but reducing the extra burden on renters.