Aberdeen endured a frustrating day in front of goal as Hearts held out for a point at Pittodrie.
Ryan Jack did not make the most of the Dons' first real chance after getting in behind the Hearts defence.
Scott Vernon then struck the visitors' crossbar as the hosts finished the first half strongly.
Hearts created little but almost snatched a dramatic victory when Andy Driver's drive hit the post deep into injury time.
John McGlynn's side preserve their unbeaten start to the Scottish Premier League, while Aberdeen remain in the hunt for their first home win of the campaign, having also drawn 0-0 with Ross County earlier in August.
Clear-cut opportunities were scarce in the first half with the creative talents of Aberdeen trio Ryan Fraser, Jonny Hayes and Jack and Hearts winger Driver finding it difficult to carve out openings.
John Sutton had a shot blocked by Dons defender Gary Naysmith and the striker was off target with a couple of other efforts as Hearts asserted themselves.
However, Aberdeen began to take control of the game as half-time approached.
Gavin Rae's pass split the Hearts defence and Jack's shot was blocked by Jamie MacDonald.
The Aberdeen fans demanded a red card after Mehdi Taouil fouled Rae in Aberdeen's inside right channel, but referee Alan Muir booked the Hearts player, deeming that Andy Webster's presence in a central position meant Rae had not been denied a clear scoring chance.
Jack then crossed for Vernon and the forward's volley on the run rattled the bar.
Marius Zaliukas was replaced with Denis Prychynenko at the break, with the Hearts captain having been carrying a thigh injury.
Jack's drive was spilled by MacDonald and Vernon, who was well placed to convert, touched the rebound onto the post. The striker's blushes were spared by the offside flag.
Naysmith limped off with what appeared to be a hamstring problem, giving Chris Clark an appearance off the bench.
A header from Vernon was well kept out by MacDonald, but the offside flag had once again been raised.
Jason Holt and then Dale Carrick were introduced for Hearts, Arvydas Novikovas and Callum Paterson withdrawn, but the visitors were struggling to get a foothold in the match with Sutton ever more isolated in attack as the second half wore on.
Aberdeen manager Craig Brown elected for a double change, with Vernon and Fraser making way for Fallon and Josh Magennis.
Fallon was soon posing a threat to the Hearts defence, having an attempt blocked by Danny Grainger and then heading over.
Hearts sustained further pressure and then had their best chance of the match with a stoppage-time break.
Ryan McGowan crossed towards Driver and the winger's right-foot shot ricocheted off the post seconds before the full-time whistle.
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