If everyone doubles their living standards over 10 years, absolute inequality rises (there is always going to be someone with no wealth/income because they want to live that way, so the absolute gap in wealth increases as the wealthiest member of society gets wealthier).
If the amount of wealth is doubled due to some center of extreme productivity (eg, a Silicon Valley) then that center is going to get preferential access to new wealth. Relative inequality rises.
In both of these scenarios, wealth inequality and living standards can raise in tandem. These are perfectly fine outcomes. Inequality is only interesting as a symptom of deeper issues affecting living standards. Should workers be rewarded according to their productivity? That would be nice, but then we would need to be very careful how we decided who got what job. There are jobs that are rare, highly productive and easy to do where the people holding them would be unfairly rewarded. That would be similar to what we have now. Focusing on median & minimum living standards will get much better results. If those are rising, inequality is not an issue.