Assume I am constructing a portfolio picking N sectors and investing in them. They, on average, it will generate an average return and I will be diversified well if N is large. The next step I could think of is to use leverage, for example 2X or 3X ETF in those sectors I chose initially. So the same portfolio but every sector is invested in via leveraged etf. Diversification would not change in this case as I am still exposed to the same set of sectors. Am I taking some additional risk? I assume there is something wrong with this strategy of levering up the returns.
Yes, diversification will change because the weighting of your portfolio will change. If you have a portfolio that's 99% in one stock and 1% in an index you are not well diversified.
Similarly if you have 50% in an ETF that is 3X weighted toward one sector and 50% in a non-leveraged market index, you are still overweight in the sector due to the leverage.
Leverage will definitely increase the risk as the risk of that portion of your portfolio will increase (unless you're using leverage to obtain a better balance, but I suspect that is not the case here).
In step 1 you decided how many sectors, and how to divide your money across each of the sectors.
In step 2 you took a subset of the sectors and decided to add leverage. You have now changed the weighting of the sectors. Those ones with leverage, may have higher returns, or they could have higher losses. If you pick poorly, you could lose money while the portfolio you setup in step one would have been fine.
The leveraged ETFs are supposed to be short term so if you were planning on holding your investments for the long term in step one, that would be another change.
Yes, leverage has risk, regardless of the underlying asset in question. With leverage, the returns per $ of your equity are magnified, and the added debt obligation creates both an ongoing cashflow issue as well as the potential to increase your total loss to your initial investment + owed debt. The diversification of your portfolio has nothing to do with this. It is a different type of risk.