The equidistribution of elliptic Dedekind sums and generalized Selberg–Kloosterman sums

We show that if

L_{1}

and

L_{2}

are linear transformations from

Z^{d}

to

Z^{d}

satisfying certain mild conditions, then, for any finite subset

A

of

Z^{d}

,

| L_{1} A + L_{2} A | \geq<#comment/> {(| det (L_{1}) |^{1 / d} + | det (L_{2}) |^{1 / d})}^{d} | A | -<#comment/> o (| A |) .

This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of

L_{1}

and

L_{2}

. As an application, we prove a lower bound for

| A + λ<#comment/> \cdot<#comment/> A |

when

A

is a finite set of real numbers and

λ<#comment/>

is an algebraic number. In particular, when

λ<#comment/>

is of the form

(p / q)^{1 / d}

for some

p, q, d \in<#comment/> N

, each taken as small as possible for such a representation, we show that

| A + λ<#comment/> \cdot<#comment/> A | \geq<#comment/> (p^{1 / d} + q^{1 / d})^{d} | A | -<#comment/> o (| A |) .

This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case

λ<#comment/> = \sqrt{2}

.