It's been 100 days since Sadiq Khan was elected as London mayor. So what has he achieved in that time?

Well if you watch Khan's promotional video released today, you'd believe it was quite a lot. In it Khan takes credit for action on air quality, equal pay, the night tube, fares, the green belt, housebuilding, policing, terrorism, and cycling.

However, a closer look at his list of claimed achievements reveals that many of them are rather more insubstantial than they first appear. Let's go through them one by one.

Tackling air quality

Khan leads off with a boast of having launched "the most ambitious plan to tackle air anywhere in the world." If that sounds familiar it's because it is. Khan's predecessor Boris Johnson also boasted of putting in place "the most ambitious and comprehensive measures in the world to tackle emissions" two years ago.

So how does Khan's own "ambitious" plans differ from Boris's? Well some of his proposals so far have been welcomed as a step forward by clean air campaigners. In particular, his proposals to expand London's Ultra Low Emission Zones were described as "hugely positive" by leading anti-pollution campaigners Client Earth.

However, for every step forward Khan has taken on this issue, he's also taken a step back. His decision to reverse his predecessor's opposition to the expansion of City Airport in East London will lead to the further deterioration in air quality in one of the most polluted parts of the capital. Similarly indications that he plans to push ahead with at least some of Johnson's controversial East London river crossings, means that there will be even more car journeys and therefore even more pollution in one of the most public-transport deprived parts of the capital.

Equal pay

Khan also boasts of a "real plan to tackle pay inequality" across the capital. However, the details of that plan so far remain elusive. Last month, he released a gender pay audit of City Hall. It found that male full-time workers at City Hall were paid 4.6% more than their female counterparts. This compares to an 11.9% pay gap among all London workers. Khan promised to achieve equal pay for men and women both at City Hall and its associated functional bodies. He has also given a large proportion of senior positions in his administration to women, in contrast to his predecessor.

While this will be welcomed by equality campaigners it's unclear what if any action Khan either can or will take to tackle the issue in the capital as a whole. So far, Khan has only said that he wants to "encourage" London employers to follow his lead. With little direct powers, this may be the most he is able to achieve. However, it hardly matches the billing given in his video. And if at the end of his four year term, the pay gap between men and women in London remains the same, or gets wider, then it will be hard for him to claim this as a major achievement.

The Night Tube

This week TfL launched trials of London's new Night Tube service. This amusingly involved TfL employees throwing cold vegetable soup onto the seats of Tube trains in an attempt to simulate the effects of thousands of inebriated Londoners using the system late on Friday and Saturday nights. On the face of it the commencement of the Night Tube will probably be remembered as a major achievement of Khan's mayoralty. However, the policy was actually instigated by Khan's predecessor, albeit without much success. The service, which was meant to launch last autumn was held up by Tube unions whose opposition to the scheme largely evaporated within days of the new Labour mayor being elected.

Fares

Sadiq Khan won the London mayoral election on a promise to freeze fares, repeatedly telling Londoners that they "won't pay a penny more for their travel in 2020 than they do today". However, shortly after entering City Hall, Khan back-pedalled on this pledge, admitting that the freeze would not extend to travelcards or journeys on non-TfL operated services. In a logic-bending move worthy of Johnson at his peak, Khan then tried to claim that he had never actually promised to freeze all fares in the first place, something which is easily disproven by one look at his manifesto. In another Boris-style move, he also blamed the broken promise on the government, saying that it was "for the DfT to make sure they fulfil a promise that I made to Londoners."

Had it come at a later stage, Khan's fares reversal could have done some serious damage. However, coming as it did during the peak of Khan's honeymoon period, he largely seems to have got away with it. Whether this will still be the case in January when thousands of Londoners realise the cost of their travelcards have gone up for yet another year, remains to be seen.

The green belt

Claims that Sadiq Khan would concrete over London's green belt were a central part of Zac Goldsmith's failed mayoral bid. Khan denied that he had any such plans and has so far held to his word. Proposals to build a new football stadium and housing in Chislehurst were approved by the Conservative-led Bromley Council earlier this year, but blocked by Khan because of potential infringement on the green belt.

Housebuilding

While suburban campaigners will no doubt welcome Khan's opposition to housebuilding on the green belt, it does little to further his central pledge to radically increase housebuilding in the city. London's housing crisis was the number one issue during the campaign and remains the central issue for most Londoners according to opinion polls.

However, those hoping for an immediate surge in housebuilding are likely to be disappointed. Under questioning from the London Assembly, Khan has already admitted that he will be unlikely to meet his own targets to increase affordable and other housebuilding in the first years of his mayoralty. In many ways this is understandable. Boris Johnson last year presided over the lowest number of affordable housing starts since the early 90s and it will obviously take the new mayor some time to turn things around. However, there is little sign so far that real action has been taken. Khan's video today claims merely that he has "started building my new 'Homes for Londoners' team to ensure there are genuinely affordable homes to rent and buy." While this is better than not starting to build his team, it hardly amounts to a major achievement of his first 100 days. Londoners will want to see something a lot more substantial on housing before the year is out.

Policing

With crime in London continuing to fall, policing is not the big issue in the capital that it once was. As a result, Khan's plans for the Met do not seem terribly ambitious. In his video, Khan boasts that he is pushing ahead with plans for "an extra dedicated police officer in every ward by 2017". The word "dedicated" is crucial here. This is more about the allocation of existing officers rather than a real increase in numbers. However, neighbourhood policing remains popular, despite the fact that evidence that "bobbies on the beat" have any real effect on crime levels is difficult to find.

Some other moves taken by Khan are more interesting. His announcement that he plans to sell Johnson's controversial water cannon, which are currently being stored at public expense in a training centre in Kent, is a welcome symbolic move away from the increasingly authoritarian direction the Met had been heading under Johnson.

Terrorism

London has so far remained remarkably free from the horrors of the kind of major terror attack seen recently in other European cities. This is largely due to the incredible preventative work of the Metropolitan police and the security services. However, there remains serious doubts about how the police could respond if a roaming attack such as those seen elsewhere in Europe did take place, particularly given the shortage of trained firearms officers in the capital. As a result Khan has launched a 'terror preparedness review' led by the respected former Metropolitan Police Authority Chair and Labour peer Toby Harris. So far, so good.

Cycling

If Johnson's time at City Hall is remembered for anything, it will most likely be for his policies on cycling. The former mayor and his cycling commissioner Andrew Gilligan took bold and controversial steps to increase cycling infrastructure despite opposition from many in his own party and Transport for London. Khan has promised to continue this agenda and has pushed ahead with some of Johnson's cycle superhighway plans. However there are at least some signs that he may not have quite the zeal of his predecessor when it comes to cycling. During the mayoral campaign, Khan signalled that future cycle superhighway lanes may not be as wide as existing ones and that physical segregation between cyclists and drivers (which cycling campaigners insist is essential) may not be permanent. Transport for London, which privately fought against some of Johnson's cycling projects, are also reportedly hoping to slash the amount of money spent on future schemes.

Of course these are still very early days and as with any new administration, the tone set by the new incumbent is as important as any concrete actions they take. On this measure, Khan has been a clear success. His principled and professional performance during the EU referendum campaign as well as his dignified and practical reaction to the result, has been impressive. His repeated insistence to EU citizens that they will remain welcome in the city also stands in stark contrast to the misjudged decision by Theresa May to use their status as a bargaining chip in Brexit negotiations. His public attacks on Donald Trump and his personal support for issues such as the gay rights movement have confirmed him as an effective leader of a multicultural and progressive city.

Perhaps the biggest success of Khan's mayoralty so far has been the lack of any obvious disasters. By the same stage in Johnson's first term, he was already mired in a series of gaffes, missteps and resignations. As a result, one poll last month found that 45% of Londoners thought he was doing well as mayor, as opposed to just 15% who thought he was doing badly.

Yet with another 40% undecided however, Londoner's final verdict on their new mayor is clearly still out. And there is good reason for them to be undecided. Khan is clearly a competent operator with real political skill. Yet in some respects Khan has benefited by comparison both with the inaction of Johnson as well as the disaster zone that is the Labour party's national leadership.

But with Londoners facing major problems on housing, inequality and the environment, it is not enough for Khan to simply rely on the inadequacies of his predecessor in order to shine by comparison. In a statement today, he insists that he is "driven by a burning ambition" and is "impatient to deliver for London". So far Khan has demonstrated plenty of ambition. He hasn't yet demonstrated a great deal of impatience. That will need to change over the coming months and years.

Adam Bienkov is the deputy editor of Politics.co.uk

The opinions in politics.co.uk's Comment and Analysis section are those of the author and are no reflection of the views of the website or its owners