This paper investigates score-based diffusion models when the underlying target distribution is concentrated on or near low-dimensional manifolds in high-dimensional space, a common feature of natural image distributions. Despite previous efforts to understand diffusion models, theoretical support for low-dimensional structures remains limited. The paper strengthens this understanding by analyzing the Denoising Diffusion Probabilistic Model (DDPM), showing that the error in each denoising step depends on the ambient dimension, which is unavoidable. A unique coefficient design is identified that achieves a convergence rate of $ O(k^2/\sqrt{T}) $, where $ k $ is the intrinsic dimension and $ T $ is the number of steps. This demonstrates that DDPM can adapt to unknown low-dimensional structures, highlighting the importance of coefficient design. The analysis uses novel tools to characterize algorithmic dynamics in a deterministic manner. The paper shows that the DDPM sampler's performance is influenced by discretization and score estimation errors. Existing results suggest that the number of steps needed for accuracy scales with the ambient dimension, but empirical evidence suggests that natural images are concentrated on low-dimensional manifolds. The paper's main contributions are showing that a particular coefficient design bounds the error of the DDPM sampler and that this design avoids discretization error proportional to the ambient dimension. This is the first theoretical demonstration that DDPM can adapt to unknown low-dimensional structures.This paper investigates score-based diffusion models when the underlying target distribution is concentrated on or near low-dimensional manifolds in high-dimensional space, a common feature of natural image distributions. Despite previous efforts to understand diffusion models, theoretical support for low-dimensional structures remains limited. The paper strengthens this understanding by analyzing the Denoising Diffusion Probabilistic Model (DDPM), showing that the error in each denoising step depends on the ambient dimension, which is unavoidable. A unique coefficient design is identified that achieves a convergence rate of $ O(k^2/\sqrt{T}) $, where $ k $ is the intrinsic dimension and $ T $ is the number of steps. This demonstrates that DDPM can adapt to unknown low-dimensional structures, highlighting the importance of coefficient design. The analysis uses novel tools to characterize algorithmic dynamics in a deterministic manner. The paper shows that the DDPM sampler's performance is influenced by discretization and score estimation errors. Existing results suggest that the number of steps needed for accuracy scales with the ambient dimension, but empirical evidence suggests that natural images are concentrated on low-dimensional manifolds. The paper's main contributions are showing that a particular coefficient design bounds the error of the DDPM sampler and that this design avoids discretization error proportional to the ambient dimension. This is the first theoretical demonstration that DDPM can adapt to unknown low-dimensional structures.