Thanks to a Georgia supreme court ruling, a week of early voting on Saturday began in nearly two dozen counties in the state for a contentious runoff between Democratic senator Raphael Warnock and Republican opponent Herschel Walker.

Recent polling commissioned by AARP shows Warnock with a four point lead over the Donald Trump-endorsed Walker ahead of the December 6 election.

The survey by Fabrizio and Associates found that Warnock had 51% of support from respondents – the first time Warnock secured a majority this year –compared to 47% for Walker. That’s higher than the 49.4% of the vote Warnock received in the initial contest on 8 November.

The poll found that Black voters and voters under 50 drove support for Warnock in particular, as well as a growing support from independents.

A week after the election, Warnock’s campaign sued Georgia over its election integrity law that restricted early voting on the Saturday after Thanksgiving. The day after Thanksgiving is also a state holiday in Georgia, originally to commemorate Robert E Lee, the Confederate civil war general.

The state’s law notes that counties may start early voting “as soon as possible” after the state certifies results from the general election, with a mandatory period from 28 November to 2 December. On Wednesday, the Georgia supreme court allowed early voting to take place.

The stakes are still high in this year’s runoff, even as Democrats managed to already win 50 Senate seats on 8 November.

For Democrats, a Warnock win would mean they would secure an outright majority in the US Senate, allowing them to hold majority control of committees and making it easier for Joe Biden’s appointees to advance.

It would also give Democratic lawmakers more security when it comes to passing legislation and allow them to rely less on adjusting to more conservative Democrats like Senators Kyrsten Sinema of Arizona and Joe Manchin of West Virginia, who have repeatedly blocked legislation.

Most recently, Manchin’s opposition played a crucial role in shaping the Inflation Reduction Act, a sweeping $739bn domestic spending and climate change package signed into law in August following negotiations. Still, Congress remains divided as Republicans wrested control of the US House.

Federal Election Commission filings show that Warnock’s campaign holds more than $29m in cash on hand ahead of the runoff, three times more than Walker does ($9.8m). The ad tracking firm AdImpact noted that Democratic-aligned groups have pumped $25m into television ads for the runoff while Republican groups spent $16m.

Notably, while other Republican allies have rallied behind Walker and as Republican groups like the Senate Leadership Fund have spent more than $10m since the general election, Trump has not announced a trip to Georgia to back Walker.

Trump’s standing within the Republican party has taken a hit since the midterm elections when Democrats performed far stronger than expected, holding the senate and restricting Republicans in the House to just a narrow majority. Trump-backed candidates in particular mostly performed poorly.

Walker deflected when asked about Trump’s endorsement, telling Fox Business: “This is not Trump’s race. This is Herschel Walker’s race.”

Barack Obama plans to travel to Atlanta next week to speak at a rally in support of Warnock. Biden has yet to announce a trip.