In attempt to deepen my understanding of Dedekind sums, I've proven the following identity $ \sum_{i = 0}^{t} \sum_{j = 0}^{b(t-i)} \left \lbrace c \left( t - i - \frac{j}{b} \right) \right \rbrace = \frac{t(t+1)}{4}(b - \gcd(b,c)), $ where $b,c,t \in \mathbb{N}$ and $\lbrace \cdot \rbrace$ denotes the fractional part function.
I'd like to compute the following more general double sums $ \sum_{i = 0}^{a t} \sum_{j = 0}^{\lfloor b(t-i/a) \rfloor} \left \lbrace c \left( t - \frac{i}{a} - \frac{j}{b} \right) \right \rbrace \qquad \text{and} \qquad \sum_{i = 1}^{a t} \sum_{j = 1}^{\lfloor b(t-i/a) \rfloor} \left \lbrace c \left( t - \frac{i}{a} - \frac{j}{b} \right) \right \rbrace \qquad a, b, c, t \in \mathbb{N}, $ where $\lfloor \cdot \rfloor$ denotes the floor function. The sums clearly vanish if $a$ and $b$ are divisors of $c$ (by the vanishing of the fractional part) and (I expect) grow as $O(t^{2})$ otherwise. Are such sums in the literature, even for some special cases of pairwise coprime coefficients $a, b$ and $c$? Any help and/or hints are certainly appreciated!
Harder Problem(s): If the above is trivial, then suppose $a, b, c$ and $t$ are real and compute the following sums $ \sum_{i = 0}^{\lfloor a t \rfloor} \left \lbrace b \left( t - \frac{i}{a} \right) \right \rbrace \qquad \text{and} \qquad \sum_{i = 0}^{\lfloor a t \rfloor} \sum_{j = 0}^{\lfloor b(t-i/a) \rfloor} \left \lbrace c \left( t - \frac{i}{a} - \frac{j}{b} \right) \right \rbrace $ as well as $ \sum_{i = 1}^{\lfloor a t \rfloor} \left \lbrace b \left( t - \frac{i}{a} \right) \right \rbrace \qquad \text{and} \qquad \sum_{i = 1}^{\lfloor a t \rfloor} \sum_{j = 1}^{\lfloor b(t-i/a) \rfloor} \left \lbrace c \left( t - \frac{i}{a} - \frac{j}{b} \right) \right \rbrace. $