We investigate the Sidon set problem in the modular case and its corresponding version in Ramsey theory. Specifically, we study the function ̅F(n) that maximizes the size of a Sidon set in Zn, as well as the minimum n such that Zn admits no Sidon r-partition (a partition whose parts are all Sidon sets), denoted by ̅SR(r), for a fixed positive integer r. We use known results and the pigeonhole principle to establish an upper bound of ̅SR(r), which allows us to find the exact values of ̅SR(r) for r ϵ {2, 3, 4, 7}. We also present a criterion for determining the non-existence of almost difference sets (ADS) in Zn. By exploiting such criterion and the connection between ADS sets and the existence of Sidon sets in Zn, we derive nontrivial upper bounds of ̅F(n) for infinitely many values of n, and we refine the upper bound on ̅SR(r) in multiple cases, determining also the exact value for r ϵ {5, 6}. Our findings shed new light on the behavior of Sidon sets in cyclic groups. In particular, we find infinitely many values for which ̅F(n) > ̅F(n + 1).

The numerical reconstruction of digitally recorded holograms suffers from speckle noise. An accelerated method that uses general-purpose computing in graphics processing units to reduce that noise is shown. The proposed methodology utilizes parallelized algorithms to record, reconstruct, and superimpose multiple uncorrelated holograms of a static scene. For the best tradeoff between reduction of the speckle noise and processing time, the method records, reconstructs, and superimposes six holograms of 1024 × 1024 pixels in 68 ms; for this case, the methodology reduces the speckle noise by 58% compared with that exhibited by a single hologram. The fully parallelized method running on a commodity graphics processing unit is one order of magnitude faster than the same technique implemented on a regular CPU using its multithreading capabilities. Experimental results are shown to validate the proposal.