Number of arithmetic progressions in dense random subsets of ℤ/nℤ

Local limit theorems for subgraph counts

“…As mentioned earlier, by transferring our result in the fixed-size model, we can also achieve a local limit theorem in the model where each element is chosen with a probability 𝑝 ∈ (0, 1), hence answering a question raised by the authors and Berkowitz [6] and providing an optimal anticoncentration result, a direction suggested by Fox, Kwan, and Sauermann [11]. We stress here that the distribution in the case where each element is chosen with probability 𝑝 is not a pointwise Gaussian as a local central limit theorem in this model is false due to the results of Berkowitz and the authors [6]. Instead the distribution is a mixture of an infinite ensemble of Gaussians, reflected by a theta series, as hypothesized in [6] and discussed further in the final section of the paper.…”

“…Now 𝜎 𝑚 = (1 + 𝑂 𝜆 (𝑛 𝜀−1∕4 ))𝜎 and 𝜇 𝑚 = 𝑓(𝑦) + 𝑂 𝜆 ((log 𝑛) 2 Substituting in this expression we find that the above, up to tolerable losses, is This allows us to answer a question of the authors and Berkowitz [6,Question 16] regarding the maximum ratio between pointwise probabilities near the mean. Indeed, Theorem 7.8 precisely pins down these probabilities to what was expected given the heuristics in [6]. The answer ultimately is the (predicted) ratio of two infinite sums as given above; explicitly, if…”

“…This example highlights the power of deducing a local limit theorem from a 'fixed size' model, especially in a case such as this where the independent model does not satisfy a local central limit theorem as demonstrated in [6]. Indeed, we end up with the 'central limit behavior' at the scale of 𝜎 𝑘 , along with a multiplier that depends on the 𝜎 𝑘 𝑛 −1∕2 scale that oscillates according to a theta series.…”

“…Finally, we note that the failure to have components 'which fluctuate independently' is ultimately the reason that a local central limit theorem does not necessarily hold for disconnected graphs (or for 𝑘-APs [6]). This argument is carried out in Section 6 and in particular the short Theorem 6.1 may serve a motivating example of why the proof isolates various 'fluctuating' components at each stage.…”

“…For the model in which each element is picked independently, however, a local central limit theorem does not hold. Recent work by the authors and Berkowitz [6] demonstrates in a quantitative sense that the distribution behaves similarly to a convolution of discrete Gaussians at two scales, but does not prove a local limit theorem. By transferring our result in the fixed-size model, we can achieve this control, hence answering a question raised in [6].…”