We consider the maximal k-spacing % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0Jd9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa% aaleaacaWGUbaabeaakiabg2da9maaxababaGaciyBaiaacggacaGG% 4baaleaacaaIWaGafyizImQba0bacaWGPbGafyizImQba0bacaWGUb% Gaey4kaSIaaGymaiabgkHiTiaadUgaaeqaaOWaaeWaaeaacaWGvbWa% aSbaaSqaaiaad6gacaWGPbGaey4kaSIaam4AaaqabaGccqGHsislca% WGvbWaaSbaaSqaaiaad6gacaWGPbaabeaaaOGaayjkaiaawMcaaaaa% !4FE9! $$M_n = \mathop {\max }\limits_{0\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } n + 1 - k} \left( {U_{ni + k} - U_{ni} } \right)$$ where U n1≦...≦U nn are the order statistics of an i.i.d. sample of size n from the uniform distribution on [0, 1], and U n0=0, U nn+1=1. The integer k is allowed to vary with n at a rate not exceeding log n. We obtain laws of the iterated logarithm for all the k's in the given range. For small k, the methods used in the proofs are borrowed from extreme value theory. For larger k, the techniques are reminiscent of those used in the proof of the Erdos-Renyi theorem.