Voters across the Sun Belt say that Donald J. Trump improved their lives when he was president — and worry that a Kamala Harris White House would not — setting the stage for an extraordinarily competitive contest in three key states, according to the latest polls from The New York Times and Siena College.

The polls found that Mr. Trump has gained a lead in Arizona and remains ahead in Georgia, two states that he lost to President Biden in 2020. But in North Carolina, which has not voted for a Democrat since 2008, Ms. Harris trails Mr. Trump by just a narrow margin.

The polls of these three states, taken from Sept. 17 to 21, presented further evidence that in a sharply divided nation, the presidential contest is shaping up to be one of the tightest in history.

[These latest Times/Siena results are some of the best results for Donald Trump in these states for weeks, Nate Cohn writes.]

Arizona, Georgia and North Carolina are on the roster of seven battleground states where the focus of both the Trump and Harris campaigns has been since Labor Day. Ms. Harris has shown relative strength in several key states across the Midwest, including, most critically to her hopes of becoming president, Pennsylvania.

But Arizona, which Mr. Biden won by just over 10,400 votes in 2020, now presents a challenge for the Harris campaign. Mr. Trump is ahead, 50 percent to 45 percent, the poll found. A Times/Siena poll there in August found Ms. Harris leading by five percentage points. Latino voters, in particular, appear to have moved away from Ms. Harris, though a significant number — 10 percent — said they were now undecided. And Mr. Trump is benefiting from ticket splitting there: While Ms. Harris is trailing, the poll shows that the Democratic candidate for Senate is ahead.