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.

In North Carolina, which Mr. Trump won by under 75,000 votes in 2020, the former president has a slim lead over Ms. Harris, drawing 49 percent of the vote compared with 47 percent for Ms. Harris. (The poll was mostly conducted before reports that Mark Robinson, the Republican candidate for governor there, had made disturbing posts in a pornography forum, which some Republicans fear could hurt Mr. Trump in the state.) And in Georgia, a state that Mr. Biden won by just under 11,800 votes in 2020, Mr. Trump continues to have a slight lead over Ms. Harris, 49 percent to 45 percent. The margin of error in each state is between four and five percentage points.

[Follow the latest polls and see updated polling averages of the Harris vs. Trump matchup.]

The polls found that voters in this part of the country were worried about their own future and the future of the nation, suggesting that Mr. Trump’s dark campaign rhetoric — “Our country is being lost, we’re a failing nation,” he said in the debate — could be resonating with some voters. A plurality said the nation’s problems were so bad that it was in danger of failing. Republicans were much more likely to hold that unsettled view of the future than Democrats, 72 percent to 16 percent.

“Whatever road we’re on right now just, to me, does not look like it’s going to end well,” said Tyler Stembridge, 41, a fire captain in Centerville, Ga., and a Republican who said he voted for Mr. Trump in 2020 and intended to support him again in November.

In one striking finding, nearly four years after Mr. Trump was impeached by the House for “incitement of insurrection” for his role in the rioting of Jan. 6, 2021, at the U.S. Capitol, respondents were evenly divided over the question over whether Mr. Trump or Ms. Harris would do a better job handling democracy.

But in one sign of how these contests remain up for grabs, about 15 percent of the electorate in Arizona, Georgia and North Carolina described themselves as undecided or not definitely decided, leaving open the possibility that they could still change their minds. This group of voters had leaned toward Ms. Harris in these states in August but now lean slightly more toward Mr. Trump.