AB system.

Data and sample selection
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The UDS K-band galaxy sample
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Our study is based on deep $K$-band imaging from the UKIRT Infrared Deep Sky Survey (UKIDSS; Lawrence et al. 2007) Ultra-Deep Survey (UDS; Almaini et al., in preparation). The UDS is the deepest of the UKIDSS surveys, covering $0.77$ square degrees in the $J$, $H$ and $K$ bands. We use the 8th UDS data release (Hartley et al. 2013), reaching depths of $J=24.9$, $H=24.2$ and $K=24.6$ (AB, 5$\sigma$, 2-arcsec apertures). The final UDS data release (June 2016) achieved estimated depths $J=25.4, H=24.8, K=25.3$, and will be used to extend our PSB studies in future work.

To complement the near-infrared imaging from UKIDSS, the UDS has deep optical coverage from [*Subaru*]{} Suprime-CAM, to depths of $B= 27.6,
~ V= 27.2, ~ R = 27.0, ~ i^{\prime} = 27.0$ and $ z^{\prime} = 26.0$ (AB, 5$\sigma$, $2$ arcsec), as described in Furusawa et al. (2008). Additional $u^{\prime}$-band imaging is provided by the Canada-France-Hawaii Telescope (CFHT) MegaCam instrument, reaching $ u^{\prime} = 26.75$ (AB, 5$\sigma$, $2$ arcsec). Deep imaging at longer near-infrared wavelengths is provided by the [*Spitzer*]{} UDS Legacy Program (SpUDS; PI: Dunlop), achieving depths of 24.2 and 24.0 