The former world No. 1 and two-time Indian Wells runner-up was deep in a third-set tiebreaker with Fils, an in-form Frenchman who had already saved two match points, when he found himself jumping for joy as a final volley sailed long to secure a 6-4, 2-6, 7-6 (7) victory.

As with Zheng, Svitolina, and even Swiatek, Indian Wells has inspired Medvedev and Fils to raise their levels of play. Neither had a good Australian Open, and neither was much better in February. But now they find themselves in the quarterfinals in the desert.