This paper introduces a new approach to achieving differential privacy by constructing a private measure from a dataset, which allows for the creation of synthetic data that maintains statistical accuracy while preserving privacy. The key innovation is the use of a "superregular random walk," a novel method that ensures the generated data remains accurate for a wide range of queries, including complex machine learning tasks like clustering and classification. The paper demonstrates that this approach provides a more efficient and accurate alternative to existing methods of generating differentially private synthetic data. The authors define a private measure on a metric space, which is a probability measure that preserves the statistical properties of the original data while ensuring privacy. They show that this measure can be used to generate synthetic data that is accurate in the Wasserstein distance, a metric that is particularly well-suited for evaluating the accuracy of data in machine learning tasks. The paper also proves an asymptotically sharp min-max result for private measures and synthetic data in general compact metric spaces, showing that the accuracy of the synthetic data depends on the geometry of the metric space. A key component of the approach is the use of a superregular random walk, which is a random walk whose joint distribution of steps is as regular as that of independent random variables, yet which deviates from the origin logarithmically slowly. This property ensures that the generated synthetic data maintains a high level of accuracy while preserving privacy. The paper also shows that this approach can be extended to general compact metric spaces, making it applicable to a wide range of data types. The authors compare their approach to existing methods of generating differentially private synthetic data, highlighting the advantages of their method in terms of accuracy and efficiency. They also discuss the implications of their results for the broader field of privacy-preserving data analysis, showing that their approach can be used to create synthetic data that is accurate for a wide range of machine learning tasks. The paper concludes with a detailed analysis of the results, showing that their approach provides a more efficient and accurate alternative to existing methods of generating differentially private synthetic data.This paper introduces a new approach to achieving differential privacy by constructing a private measure from a dataset, which allows for the creation of synthetic data that maintains statistical accuracy while preserving privacy. The key innovation is the use of a "superregular random walk," a novel method that ensures the generated data remains accurate for a wide range of queries, including complex machine learning tasks like clustering and classification. The paper demonstrates that this approach provides a more efficient and accurate alternative to existing methods of generating differentially private synthetic data. The authors define a private measure on a metric space, which is a probability measure that preserves the statistical properties of the original data while ensuring privacy. They show that this measure can be used to generate synthetic data that is accurate in the Wasserstein distance, a metric that is particularly well-suited for evaluating the accuracy of data in machine learning tasks. The paper also proves an asymptotically sharp min-max result for private measures and synthetic data in general compact metric spaces, showing that the accuracy of the synthetic data depends on the geometry of the metric space. A key component of the approach is the use of a superregular random walk, which is a random walk whose joint distribution of steps is as regular as that of independent random variables, yet which deviates from the origin logarithmically slowly. This property ensures that the generated synthetic data maintains a high level of accuracy while preserving privacy. The paper also shows that this approach can be extended to general compact metric spaces, making it applicable to a wide range of data types. The authors compare their approach to existing methods of generating differentially private synthetic data, highlighting the advantages of their method in terms of accuracy and efficiency. They also discuss the implications of their results for the broader field of privacy-preserving data analysis, showing that their approach can be used to create synthetic data that is accurate for a wide range of machine learning tasks. The paper concludes with a detailed analysis of the results, showing that their approach provides a more efficient and accurate alternative to existing methods of generating differentially private synthetic data.