SHANGHAI, Sept 26 (Reuters) - China and Hong Kong stocks fell on Tuesday as lingering economic worries and geopolitical tensions weighed on sentiment, with thin trading ahead of China's National Day holiday.

** China's bluechip CSI300 Index lost 0.5% by the lunch break, while the Shanghai Composite Index declined 0.3%. Hong Kong's Hang Seng Index was down 0.9%.

** The recent improvement in some economic activity indices has convinced many market participants that China's economy has already bottomed out, but "we remain cautious," Nomura said in a note.

** Many private developers in lower-tier cities are still being trapped, rise in prices of imported commodities and energy was not passed through to downstream sectors, and "geopolitical tensions have not really improved, if they have not deteriorated," Nomura said.

** U.S. President Joe Biden's administration on Monday imposed new trade restrictions on 11 Chinese and five Russian companies, accusing some of supplying components to make drones for Russia's war effort in Ukraine.

** Latest development at developers China Evergrande Group and China Oceanwide Holdings s reversed a brief respite for China's property sector, which accounts for roughly a quarter of its economy.

** An index tracking Hong Kong-listed mainland developers dropped 1.3%.

** Most sectors dropped in China. The banking sector slid 0.4% and the Defence Index lost 0.7%. (Reporting by Shanghai Newsroom; Editing by Rashmi Aich)